Amalie Emmy Noether
[ Emmy is the '']Rufname
Personal names in German-speaking Europe consist of one or several given names (''Vorname'', plural ''Vornamen'') and a surname (''Nachname, Familienname'').
The ''Vorname'' is usually gender-specific. A name is usually cited in the " Western or ...
'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noether'' (1907/08, NR. 2988); reproduced in: ''Emmy Noether, Gesammelte Abhandlungen – Collected Papers,'' ed. N. Jacobson 1983; online facsimile a
physikerinnen.de/noetherlebenslauf.html
). Sometimes ''Emmy'' is mistakenly reported as a short form for ''Amalie'', or misreported as "Emily". e.g. (, ; ; 23 March 1882 – 14 April 1935) was a
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
who made many important contributions to
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
. She discovered Noether's
First
First or 1st is the ordinal form of the number one (#1).
First or 1st may also refer to:
*World record, specifically the first instance of a particular achievement
Arts and media Music
* 1$T, American rapper, singer-songwriter, DJ, and rec ...
and
Second Theorem, which are fundamental in
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
. She was described by
Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
,
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
,
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...
,
Hermann Weyl and
Norbert Wiener as the most important
woman in the history of mathematics.
As one of the leading mathematicians of her time, she developed some theories of
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
,
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
, and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
. In physics,
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
explains the connection between
symmetry and
conservation laws
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
.
Noether was born to a
Jewish family in the
Franconia
Franconia (german: Franken, ; Franconian dialect: ''Franggn'' ; bar, Frankn) is a region of Germany, characterised by its culture and Franconian languages, Franconian dialect (German: ''Fränkisch'').
The three Regierungsbezirk, administrative ...
n town of
Erlangen
Erlangen (; East Franconian: ''Erlang'', Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative district Erlangen), and with 116,062 inhab ...
; her father was the mathematician
Max Noether
Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the f ...
. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the
University of Erlangen
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
, where her father lectured. After completing her doctorate in 1907
[Ogilvie, M. B., & Harvey, J. D. (2000). The biographical dictionary of women in science: Pioneering lives from ancient times to the mid-20th century. New York: Routledge. p. 949] under the supervision of
Paul Gordan
__NOTOC__
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor ...
, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by
David Hilbert and
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
to join the mathematics department at the
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her
habilitation was approved in 1919, allowing her to obtain the rank of ''
Privatdozent''.
Noether remained a leading member of the
Göttingen
Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911.
General information
The ori ...
mathematics department until 1933; her students were sometimes called the "Noether boys". In 1924, Dutch mathematician
B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, ''
Moderne Algebra
''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', th ...
''. By the time of her plenary address at the 1932
International Congress of Mathematicians in
Zürich
, neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon
, twintowns = Kunming, San Francisco
Zürich ...
, her algebraic acumen was recognized around the world. The following year, Germany's Nazi government
dismissed Jews from university positions, and Noether moved to the United States to take up a position at
Bryn Mawr College
Bryn Mawr College ( ; Welsh: ) is a women's liberal arts college in Bryn Mawr, Pennsylvania. Founded as a Quaker institution in 1885, Bryn Mawr is one of the Seven Sister colleges, a group of elite, historically women's colleges in the United ...
in
Pennsylvania
Pennsylvania (; ( Pennsylvania Dutch: )), officially the Commonwealth of Pennsylvania, is a state spanning the Mid-Atlantic, Northeastern, Appalachian, and Great Lakes regions of the United States. It borders Delaware to its southeast, ...
where she taught, among others, doctoral and post-graduate women including
Marie Johanna Weiss, Ruth Stauffer, Grace Shover Quinn and
Olga Taussky-Todd
Olga Taussky-Todd (August 30, 1906, Olomouc, Austria-Hungary (present-day Olomouc, Czech Republic) – October 7, 1995, Pasadena, California) was an Austrian and later Czech-American mathematician. She published more than 300 research papers on ...
. At the same time, she lectured and performed research at the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
in
Princeton, New Jersey
Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...
.
Noether's mathematical work has been divided into three "epochs".
In the first (1908–1919), she made contributions to the theories of
algebraic invariant
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descrip ...
s and
number fields
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
. Her work on differential invariants in the
calculus of variations,
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of
bstractalgebra".
In her classic 1921 paper ''Idealtheorie in Ringbereichen'' (''Theory of Ideals in Ring Domains''), Noether developed the theory of
ideals in
commutative rings into a tool with wide-ranging applications. She made elegant use of the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
, and objects satisfying it are named ''
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
'' in her honor. In the third epoch (1927–1935), she published works on
noncommutative algebra
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s and
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...
s and united the
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
with the theory of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
Personal life
![NoetherFamily MFO3120](https://upload.wikimedia.org/wikipedia/commons/0/0b/NoetherFamily_MFO3120.jpg)
Emmy Noether was born on 23 March 1882, the first of four children of mathematician
Max Noether
Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the f ...
and Ida Amalia Kaufmann, both from Jewish merchant families. Her first name was "Amalie", after her mother and paternal grandmother, but she began using her middle name at a young age, and she invariably used the name "Emmy Noether" in her adult life and her publications.
[
In her youth, Noether did not stand out academically although she was known for being clever and friendly. She was ]near-sighted
Near-sightedness, also known as myopia and short-sightedness, is an eye disease where light focuses in front of, instead of on, the retina. As a result, distant objects appear blurry while close objects appear normal. Other symptoms may include ...
and talked with a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at that early age. She was taught to cook and clean, as were most girls of the time, and she took piano lessons. She pursued none of these activities with passion, although she loved to dance.
She had three younger brothers: the eldest, Alfred, was born in 1883, was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether
Fritz Alexander Ernst Noether (7 October 1884 – 10 September 1941) was a Jewish German mathematician who emigrated from Nazi Germany to the Soviet Union. He was later executed by the NKVD.
Biography
Fritz Noether's father Max Noethe ...
, born in 1884, is remembered for his academic accomplishments. After studying in Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the third-largest city in Germany, after Berlin and Ha ...
he made a reputation for himself in applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
. He was executed in the Soviet Union in 1941. The youngest, Gustav Robert, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.
In 1935, Noether underwent surgery for an ovarian cyst
An ovarian cyst is a fluid-filled sac within the ovary. Often they cause no symptoms. Occasionally they may produce bloating, lower abdominal pain, or lower back pain. The majority of cysts are harmless. If the cyst either breaks open or causes ...
and, despite signs of a recovery, died four days later at the age of 53.
University life and education
Noether showed early proficiency in French and English. In the spring of 1900, she took the examination for teachers of these languages and received an overall score of ''sehr gut'' (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
.
This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing mixed-sex education would "overthrow all academic order". One of only two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903 she passed the graduation exam at a ''Realgymnasium
''Gymnasium'' (; German plural: ''Gymnasien''), in the German education system, is the most advanced and highest of the three types of German secondary schools, the others being ''Hauptschule'' (lowest) and ''Realschule'' (middle). ''Gymnas ...
'' in Nuremberg
Nuremberg ( ; german: link=no, Nürnberg ; in the local East Franconian dialect: ''Nämberch'' ) is the second-largest city of the German state of Bavaria after its capital Munich, and its 518,370 (2019) inhabitants make it the 14th-largest ...
.
During the 1903–1904 winter semester, she studied at the University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
, attending lectures given by astronomer Karl Schwarzschild
Karl Schwarzschild (; 9 October 1873 – 11 May 1916) was a German physicist and astronomer.
Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-r ...
and mathematicians Hermann Minkowski, Otto Blumenthal
Ludwig Otto Blumenthal (20 July 1876 – 12 November 1944) was a German mathematician and professor at RWTH Aachen University.
Biography
He was born in Frankfurt, Hesse-Nassau. A student of David Hilbert, Blumenthal was an editor of '' Mathema ...
, Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
, and David Hilbert. Soon thereafter, restrictions on women's participation in that university were rescinded.
Noether returned to Erlangen. She officially reentered the university in October 1904, and declared her intention to focus solely on mathematics. Under the supervision of Paul Gordan
__NOTOC__
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor ...
she wrote her dissertation, ''Über die Bildung des Formensystems der ternären biquadratischen Form'' (''On Complete Systems of Invariants for Ternary Biquadratic Forms'', 1907). Gordan was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert. Although it had been well received, Noether later described her thesis and a number of subsequent similar papers she produced as "crap".[ write that she completed her doctorate at Göttingen, but this appears to be an error.]
Teaching period
University of Erlangen
For the next seven years (1908–1915) she taught at the University of Erlangen's Mathematical Institute without pay, occasionally substituting for her father when he was too ill to lecture. In 1910 and 1911 she published an extension of her thesis work from three variables to ''n'' variables.
Gordan retired in the spring of 1910, but continued to teach occasionally with his successor, Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...
, who left shortly afterward for a position in Breslau. Gordan retired from teaching altogether in 1911 when Schmidt's successor Ernst Fischer arrived; Gordan died a year later in December 1912.
According to Hermann Weyl, Fischer was an important influence on Noether, in particular by introducing her to the work of David Hilbert. From 1913 to 1916 Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
of rational functions and the invariants of finite groups. This phase marks the beginning of her engagement with abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, the field of mathematics to which she would make groundbreaking contributions.
Noether and Fischer shared lively enjoyment of mathematics and would often discuss lectures long after they were over; Noether is known to have sent postcards to Fischer continuing her train of mathematical thoughts.
University of Göttingen
In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
. Their effort to recruit her, however, was blocked by the philologists
Philology () is the study of language in oral and written historical sources; it is the intersection of textual criticism, literary criticism, history, and linguistics (with especially strong ties to etymology). Philology is also defined ...
and historian
A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human race; as well as the st ...
s among the philosophical faculty: Women, they insisted, should not become '' privatdozenten''. One faculty member protested: "''What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?''" Hilbert responded with indignation, stating, "''I do not see that the sex of the candidate is an argument against her admission as privatdozent. After all, we are a university, not a bathhouse.''"
Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death is unknown. At about the same time Noether's father retired and her brother joined the German Army to serve in World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
. She returned to Erlangen for several weeks, mostly to care for her aging father.
During her first years teaching at Göttingen she did not have an official position and was not paid; her family paid for her room and board and supported her academic work. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance".
Soon after arriving at Göttingen, however, she demonstrated her capabilities by proving the theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
now known as Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
, which shows that a conservation law is associated with any differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
symmetry of a physical system. The paper was presented by a colleague, F. Klein, on 26 July 1918 to a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society. American physicists Leon M. Lederman
Leon Max Lederman (July 15, 1922 – October 3, 2018) was an American experimental physicist who received the Nobel Prize in Physics in 1988, along with Melvin Schwartz and Jack Steinberger, for research on neutrinos. He also received the Wolf P ...
and Christopher T. Hill argue in their book ''Symmetry and the Beautiful Universe'' that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem".
When World War I ended, the German Revolution of 1918–1919 brought a significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed Noether to proceed with her '' habilitation'' (eligibility for tenure). Her oral examination was held in late May, and she successfully delivered her ''habilitation'' lecture in June 1919.
Three years later she received a letter from , the Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
n Minister for Science, Art, and Public Education, in which he conferred on her the title of ''nicht beamteter ausserordentlicher Professor
Academic ranks in Germany are the titles, relative importance and power of professors, researchers, and administrative personnel held in academia.
Overview
Appointment grades
* (Pay grade: ''W3'' or ''W2'')
* (''W3'')
* (''W2'')
* (''W2'', ...
'' (an untenured professor with limited internal administrative rights and functions). This was an unpaid "extraordinary" professor
Professor (commonly abbreviated as Prof.) is an academic rank at universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who professes". Professors ...
ship, not the higher "ordinary" professorship, which was a civil-service position. Although it recognized the importance of her work, the position still provided no salary. Noether was not paid for her lectures until she was appointed to the special position of ''Lehrbeauftragte für Algebra'' a year later.
Work in abstract algebra
Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she is best remembered for her contributions to abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
. In his introduction to Noether's ''Collected Papers'', Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
Biography
Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...
wrote thatThe development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.
She sometimes allowed her colleagues and students to receive credit for her ideas, helping them develop their careers at the expense of her own.
Noether's work in algebra began in 1920. In collaboration with W. Schmeidler, she then published a paper about the theory of ideals in which they defined left and right ideals in a ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
The following year she published a paper called ''Idealtheorie in Ringbereichen'', analyzing ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
s with regard to (mathematical) ideals. Noted algebraist Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
called this work "revolutionary"; the publication gave rise to the term "Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
" and the naming of several other mathematical objects as ''Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
''.
In 1924 a young Dutch mathematician, B.L. van der Waerden, arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison". In 1931 he published ''Moderne Algebra'', a central text in the field; its second volume borrowed heavily from Noether's work. Although Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin and E. Noether".
Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which became a major hub of mathematical and physical research. From 1926 to 1930 Russian topologist
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
lectured at the university, and he and Noether quickly became good friends. He began referring to her as ''der Noether'', using the masculine German article as a term of endearment to show his respect. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship from the Rockefeller Foundation. They met regularly and enjoyed discussions about the intersections of algebra and topology. In his 1935 memorial address, Alexandrov named Emmy Noether "the greatest woman mathematician of all time".
Graduate students and influential lectures
In addition to her mathematical insight, Noether was respected for her consideration of others. Although she sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to name her "a severe critic", but she combined this demand for accuracy with a nurturing attitude. A colleague later described her this way:Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all.
Göttingen
In Göttingen, Noether supervised more than a dozen doctoral students; her first was Grete Hermann
Grete Hermann (2 March 1901 – 15 April 1984) was a German mathematician and philosopher noted for her work in mathematics, physics, philosophy and education. She is noted for her early philosophical work on the foundations of quantum mechanics, ...
, who defended her dissertation in February 1925. She later spoke reverently of her "dissertation-mother". Noether also supervised Max Deuring
Max Deuring (9 December 1907 – 20 December 1984) was a German mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p. He worked also in analytic number theory.
Deuring graduated fr ...
, who distinguished himself as an undergraduate and went on to contribute to the field of arithmetic geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
; Hans Fitting, remembered for Fitting's theorem
Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows:
:If ''M'' and ''N'' are nilpotent normal subgroups of a group ''G'', then their product ''MN'' is also a nilpotent normal subgroup of ''G''; if, mor ...
and the Fitting lemma; and Zeng Jiongzhi (also rendered "Chiungtze C. Tsen" in English), who proved Tsen's theorem. She also worked closely with Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
, who greatly advanced commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
with his ''Hauptidealsatz'' and his dimension theory
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
for commutative rings.
Her frugal lifestyle at first was due to her being denied pay for her work; however, even after the university began paying her a small salary in 1923, she continued to live a simple and modest life. She was paid more generously later in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether.
Biographers suggest that she was mostly unconcerned about appearance and manners, focusing on her studies. A distinguished algebraist Olga Taussky-Todd
Olga Taussky-Todd (August 30, 1906, Olomouc, Austria-Hungary (present-day Olomouc, Czech Republic) – October 7, 1995, Pasadena, California) was an Austrian and later Czech-American mathematician. She published more than 300 research papers on ...
described a luncheon during which Noether, wholly engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed". Appearance-conscious students cringed as she retrieved the handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematical discussion she was having with other students.
According to van der Waerden's obituary of Emmy Noether, she did not follow a lesson plan for her lectures, which frustrated some students. Instead, she used her lectures as a spontaneous discussion time with her students, to think through and clarify important problems in mathematics. Some of her most important results were developed in these lectures, and the lecture notes of her students formed the basis for several important textbooks, such as those of van der Waerden and Deuring.
Several of her colleagues attended her lectures, and she allowed some of her ideas, such as the crossed product
In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product
is a basic method of constructing a new von Neumann algebra from
a von Neumann algebra acted on by a group. It is related to
the semidirect product ...
(''verschränktes Produkt'' in German) of associative algebras, to be published by others. Noether was recorded as having given at least five semester-long courses at Göttingen:[Scharlau, W. "Emmy Noether's Contributions to the Theory of Algebras" in .]
* Winter 1924/1925: ''Gruppentheorie und hyperkomplexe Zahlen'' 'Group Theory and Hypercomplex Numbers''* Winter 1927/1928: ''Hyperkomplexe Grössen und Darstellungstheorie'' 'Hypercomplex Quantities and Representation Theory''* Summer 1928: ''Nichtkommutative Algebra'' 'Noncommutative Algebra''* Summer 1929: ''Nichtkommutative Arithmetik'' 'Noncommutative Arithmetic''* Winter 1929/30: ''Algebra der hyperkomplexen Grössen'' 'Algebra of Hypercomplex Quantities''
These courses often preceded major publications on the same subjects.
Noether spoke quicklyreflecting the speed of her thoughts, many saidand demanded great concentration from her students. Students who disliked her style often felt alienated. Some pupils felt that she relied too much on spontaneous discussions. Her most dedicated students, however, relished the enthusiasm with which she approached mathematics, especially since her lectures often built on earlier work they had done together.
She developed a close circle of colleagues and students who thought along similar lines and tended to exclude those who did not. "Outsiders" who occasionally visited Noether's lectures usually spent only 30 minutes in the room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out."
Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and lectured at a local coffee house. Later, after Nazi Germany
Nazi Germany (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") (officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945) was ...
dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.
Moscow
In the winter of 1928–1929 Noether accepted an invitation to Moscow State University
M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
, where she continued working with P.S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. She worked with the topologists Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
and Nikolai Chebotaryov
Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev, uk, Мико́ла Григо́рович Чеботарьо́в, russian: Никола́й Григо́рьевич Чеботарёв) ( – 2 July 1947) was a Ukrainia ...
, who later praised her contributions to the development of Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.
Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution. She was especially happy to see Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik
The Bolsheviks (russian: Большевики́, from большинство́ ''bol'shinstvó'', 'majority'),; derived from ''bol'shinstvó'' (большинство́), "majority", literally meaning "one of the majority". also known in English ...
project. This attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess".
Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933 he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry. Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union. Meanwhile, her brother Fritz
Fritz originated as a German nickname for Friedrich, or Frederick (''Der Alte Fritz'', and ''Stary Fryc'' were common nicknames for King Frederick II of Prussia and Frederick III, German Emperor) as well as for similar names including Fridolin a ...
accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk
Tomsk ( rus, Томск, p=tomsk, sty, Түң-тора) is a city and the administrative center of Tomsk Oblast in Russia, located on the Tom River. Population:
Founded in 1604, Tomsk is one of the oldest cities in Siberia. The city is a n ...
, in the Siberian Federal District of Russia, after losing his job in Germany, and was subsequently executed during the Great Purge
The Great Purge or the Great Terror (russian: Большой террор), also known as the Year of '37 (russian: 37-й год, translit=Tridtsat sedmoi god, label=none) and the Yezhovshchina ('period of Yezhov'), was Soviet General Secret ...
.
Recognition
In 1932 Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics. The prize included a monetary reward of and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to the Göttingen ''Gesellschaft der Wissenschaften'' (academy of sciences) and was never promoted to the position of ''Ordentlicher Professor
Academic ranks in Germany are the titles, relative importance and power of professors, researchers, and administrative personnel held in academia.
Overview
Appointment grades
* (Pay grade: ''W3'' or ''W2'')
* (''W3'')
* (''W2'')
* (''W2'', ...
'' (full professor).
Noether's colleagues celebrated her fiftieth birthday in 1932, in typical mathematicians' style. Helmut Hasse dedicated an article to her in the '' Mathematische Annalen'', wherein he confirmed her suspicion that some aspects of noncommutative algebra
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
are simpler than those of commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, by proving a noncommutative reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...
. This pleased her immensely. He also sent her a mathematical riddle, which he called the "mμν-riddle of syllables". She solved it immediately, but the riddle has been lost.
In September of the same year, Noether delivered a plenary address (''großer Vortrag'') on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich
, neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon
, twintowns = Kunming, San Francisco
Zürich ...
. The congress was attended by 800 people, including Noether's colleagues Hermann Weyl, Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopol ...
, and Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
. There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.
Expulsion from Göttingen by Nazi Germany
When Adolf Hitler
Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Nazi Germany, Germany from 1933 until Death of Adolf Hitler, his death in 1945. Adolf Hitler's rise to power, He rose to power as the le ...
became the German ''Reichskanzler'' in January 1933, Nazi
Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right totalitarian political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in ...
activity around the country increased dramatically. At the University of Göttingen the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by a '' privatdozent'' named Werner Weber, a former student of Noether. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."[.]
One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen." Several of Noether's colleagues, including Max Born and Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
, also had their positions revoked.
Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later wrote that "Emmy Noether—her courage, her frankness, her unconcern about her own fate, her conciliatory spirit—was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace." Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
. When one of her students appeared in the uniform of the Nazi paramilitary organization '' Sturmabteilung'' (SA), she showed no sign of agitation and, reportedly, even laughed about it later. This, however, was before the bloody events of Kristallnacht
() or the Night of Broken Glass, also called the November pogrom(s) (german: Novemberpogrome, ), was a pogrom against Jews carried out by the Nazi Party's (SA) paramilitary and (SS) paramilitary forces along with some participation fro ...
in 1938, and their praise from Propaganda Minister Joseph Goebbels.
Refuge at Bryn Mawr and Princeton, in the United States
As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them. Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
and Hermann Weyl were appointed by the Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
in Princeton
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...
, while others worked to find a sponsor required for legal immigration
Immigration is the international movement of people to a destination country of which they are not natives or where they do not possess citizenship in order to settle as permanent residents or naturalized citizens. Commuters, tourists, a ...
. Noether was contacted by representatives of two educational institutions: Bryn Mawr College
Bryn Mawr College ( ; Welsh: ) is a women's liberal arts college in Bryn Mawr, Pennsylvania. Founded as a Quaker institution in 1885, Bryn Mawr is one of the Seven Sister colleges, a group of elite, historically women's colleges in the United ...
, in the United States, and Somerville College
Somerville College, a constituent college of the University of Oxford in England, was founded in 1879 as Somerville Hall, one of its first two women's colleges. Among its alumnae have been Margaret Thatcher, Indira Gandhi, Dorothy Hodgkin, ...
at the University of Oxford
, mottoeng = The Lord is my light
, established =
, endowment = £6.1 billion (including colleges) (2019)
, budget = £2.145 billion (2019–20)
, chancellor ...
, in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.
At Bryn Mawr, Noether met and befriended Anna Wheeler, who had studied at Göttingen just before Noether arrived there. Another source of support at the college was the Bryn Mawr president, Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr. Noether in action!" Noether and a small team of students worked quickly through van der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics.
Biography
Education and early career
Van der Waerden learned advanced mathematics at the University of Amsterd ...
's 1930 book ''Moderne Algebra I'' and parts of Erich Hecke
Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms.
Biography
Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He o ...
's ''Theorie der algebraischen Zahlen'' (''Theory of algebraic numbers'').
In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner
Abraham Flexner (November 13, 1866 – September 21, 1959) was an American educator, best known for his role in the 20th century reform of medical and higher education in the United States and Canada.
After founding and directing a college-prep ...
and Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was lon ...
. She also worked with and supervised Abraham Albert and Harry Vandiver
Harry Schultz Vandiver (21 October 1882 – 9 January 1973) was an American mathematician, known for work in number theory.
He was born in Philadelphia, Pennsylvania to John Lyon and Ida Frances (Everett) Vandiver. He did not complete a form ...
. However, she remarked about Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
that she was not welcome at "the men's university, where nothing female is admitted".
Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects. In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother Fritz
Fritz originated as a German nickname for Friedrich, or Frederick (''Der Alte Fritz'', and ''Stary Fryc'' were common nicknames for King Frederick II of Prussia and Frederick III, German Emperor) as well as for similar names including Fridolin a ...
before he left for Tomsk. Although many of her former colleagues had been forced out of the universities, she was able to use the library as a "foreign scholar". Without incident, Noether returned to the United States and her studies at Bryn Mawr.
Death
In April 1935 doctors discovered a tumor
A neoplasm () is a type of abnormal and excessive growth of tissue. The process that occurs to form or produce a neoplasm is called neoplasia. The growth of a neoplasm is uncoordinated with that of the normal surrounding tissue, and persists ...
in Noether's pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst
An ovarian cyst is a fluid-filled sac within the ovary. Often they cause no symptoms. Occasionally they may produce bloating, lower abdominal pain, or lower back pain. The majority of cysts are harmless. If the cyst either breaks open or causes ...
"the size of a large cantaloupe
The cantaloupe, rockmelon (Australia and New Zealand, although cantaloupe is used in some states of Australia), sweet melon, or spanspek (Southern Africa) is a melon that is a variety of the muskmelon species (''Cucumis melo'') from the fami ...
". Two smaller tumors in her uterus
The uterus (from Latin ''uterus'', plural ''uteri'') or womb () is the organ in the reproductive system of most female mammals, including humans that accommodates the embryonic and fetal development of one or more embryos until birth. The ...
appeared to be benign and were not removed, to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse
Shock is the state of insufficient blood flow to the tissues of the body as a result of problems with the circulatory system. Initial symptoms of shock may include weakness, fast heart rate, fast breathing, sweating, anxiety, and increased thi ...
on the fourth. On 14 April she fell unconscious, her temperature soared to , and she died. " is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located."
A few days after Noether's death her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
traveled from Princeton and spoke with Wheeler and Taussky about their departed colleague. In the months that followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.
Contributions to mathematics and physics
Noether's work in abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
and topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
was influential in mathematics, while in physics, Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
has consequences for theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
and dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s. She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways. Her friend and colleague Hermann Weyl described her scholarly output in three epochs:
In the first epoch (1907–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan
__NOTOC__
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor ...
. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. After moving to Göttingen in 1915, she produced her work for physics, the two Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
s.
In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings.
In the third epoch (1927–1935), Noether focused on noncommutative algebra
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
, linear transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, and commutative number fields.
Although the results of Noether's first epoch were impressive and useful, her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B.L. van der Waerden in their obituaries of her.
In these epochs, she was not merely applying ideas and methods of earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory of ideals in rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, generalizing earlier work of Richard Dedekind. She is also renowned for developing ascending chain conditions, a simple finiteness condition that yielded powerful results in her hands. Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as elimination theory
Elimination may refer to:
Science and medicine
*Elimination reaction, an organic reaction in which two functional groups split to form an organic product
*Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
and the algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
that had been studied by her father.
Historical context
In the century from 1832 to Noether's death in 1935, the field of mathematics – specifically algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
– underwent a profound revolution, whose reverberations are still being felt. Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
s, as well as on the related problem of constructing regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s using compass and straightedge. Beginning with Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
's 1832 proof that prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s such as five can be factored in Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s, Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...
's discovery of quaternions in 1843, and Arthur Cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this new field, abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
.[G.E. .]
Background on abstract algebra and ''begriffliche Mathematik'' (conceptual mathematics)
Two of the most basic objects in abstract algebra are groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
and rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
A ''group'' consists of a set of elements and a single operation which combines a first and a second element and returns a third. The operation must satisfy certain constraints for it to determine a group: It must be closed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
(an element which, when combined with another element using the operation, results in the original element, such as adding zero to a number or multiplying it by one), and for every element there must be an inverse element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
.
A ''ring'' likewise, has a set of elements, but now has ''two'' operations. The first operation must make the set a commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
group, and the second operation is associative and distributive with respect to the first operation. It may or may not be commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
; this means that the result of applying the operation to a first and a second element is the same as to the second and first – the order of the elements does not matter. If every non-zero element has a multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/ ...
(an element ''x'' such that ''a x'' = ''x a'' = 1 ), the ring is called a ''division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
''. A ''field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
'' is defined as a commutative division ring.
Groups are frequently studied through ''group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s''. In their most general form, these consist of a choice of group, a set, and an ''action'' of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. Most often, the set is a vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, and the group represents symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. This is a type of symmetry of space, because space itself does not change when it is rotated even though the positions of objects in it do. Noether used these sorts of symmetries in her work on invariants in physics.
A powerful way of studying rings is through their ''modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
''. A module consists of a choice of ring, another set, usually distinct from the underlying set of the ring and called the underlying set of the module, an operation on pairs of elements of the underlying set of the module, and an operation which takes an element of the ring and an element of the module and returns an element of the module.
The underlying set of the module and its operation must form a group. A module is a ring-theoretic version of a group representation: Ignoring the second ring operation and the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. An important special case of this is an ''algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
''. (The word algebra means both a subject within mathematics as well as an object studied in the subject of algebra.) An algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. This operation makes the second ring into a module over the first. Often the first ring is a field.
Words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For example, the elements might be computer data words, where the first combining operation is exclusive or
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
and the second is logical conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...
. Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, her student van der Waerden recalled that
This is the ''begriffliche Mathematik'' (purely conceptual mathematics) that was characteristic of Noether. This style of mathematics was consequently adopted by other mathematicians, especially in the (then new) field of abstract algebra.
Example: Integers as a ring
The integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. Any pair of integers can be added or multiplied, always resulting in another integer, and the first operation, addition, is commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, i.e., for any elements ''a'' and ''b'' in the ring, ''a'' + ''b'' = ''b'' + ''a''. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that ''a'' combined with ''b'' might be different from ''b'' combined with ''a''. Examples of noncommutative rings include matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and quaternions. The integers do not form a division ring, because the second operation cannot always be inverted; there is no integer ''a'' such that 3 × ''a'' = 1.
The integers have additional properties which do not generalize to all commutative rings. An important example is the fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
, which says that every positive integer can be factored uniquely into prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called the ''Lasker–Noether theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are relate ...
'', for the ideals of many rings. Much of Noether's work lay in determining what properties ''do'' hold for all rings, in devising novel analogs of the old integer theorems, and in determining the minimal set of assumptions required to yield certain properties of rings.
First epoch (1908–1919): Algebraic invariant theory
Much of Noether's work in the first epoch of her career was associated with invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of transformations. As an everyday example, if a rigid yardstick is rotated, the coordinates (''x''1, ''y''1, ''z''1) and (''x''2, ''y''2, ''z''2) of its endpoints change, but its length ''L'' given by the formula ''L''2 = Δ''x''2 + Δ''y''2 + Δ''z''2 remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
's Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
, according to which different types of geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
should be characterized by their invariants under transformations, e.g., the cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, th ...
of projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
.
An example of an ''invariant'' is the discriminant ''B''2 − 4 ''A C'' of a binary quadratic form x·A x + y·B x + y·C y , where x and y are vectors and "·" is the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
or "inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
" for the vectors. A, B, and C are linear operators on the vectors – typically matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.
The discriminant is called "invariant" because it is not changed by linear substitutions ''x'' → ''a'' ''x'' + ''b'' ''y'', ''y'' → ''c'' x + ''d'' y with determinant ''a'' ''d'' − ''b'' ''c'' = 1 . These substitutions form the special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
''SL''2.[There are no invariants under the ]general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of all invertible linear transformations because these transformations can be multiplication by a scaling factor. To remedy this, classical invariant theory also considered ''relative invariants'', which were forms invariant up to a scale factor.
One can ask for all polynomials in A, B, and C that are unchanged by the action of ''SL''2; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant.
More generally, one can ask for the invariants of homogeneous polynomials A0 ''x''r ''y''0 + ... + Ar x0 ''y''r of higher degree, which will be certain polynomials in the coefficients A0, ..., Ar, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.
One of the main goals of invariant theory was to solve the "''finite basis problem''". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called ''generators'', and then, adding or multiplying the generators together. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.
Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables. He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables. Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.
First epoch (1908–1919): Galois theory
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
concerns transformations of number fields
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
that permute the roots of an equation. Consider a polynomial equation of a variable ''x'' of degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
''n'', in which the coefficients are drawn from some ground field In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion.
Use
It is used in various areas of algebra:
In linear algebra
In linear algebra, the concept of a vector space may be developed over any field.
In algeb ...
, which might be, for example, the field of real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, or the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s modulo 7. There may or may not be choices of ''x'', which make this polynomial evaluate to zero. Such choices, if they exist, are called roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
. If the polynomial is ''x''2 + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of ''x'' makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.
Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +''i'' and −''i'', where ''i'' is the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, that is, ''i'' 2 = −1 . More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
of the polynomial.
The Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial. (In mathematical jargon, these transformations are called automorphisms.) The Galois group of consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends +''i'' to −''i''. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a permutation of the ''n'' roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In its most basi ...
, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of the Galois group.
In 1918, Noether published a paper on the inverse Galois problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved.
There ...
. Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to " Noether's problem", which asks whether the fixed field of a subgroup ''G'' of the permutation group ''S''''n'' acting on the field ''k''(''x''1, ... , ''x''''n'') always is a pure transcendental extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the field ''k''. (She first mentioned this problem in a 1913 paper, where she attributed the problem to her colleague Fischer
Fischer is a German occupational surname, meaning fisherman. The name Fischer is the fourth most common German surname. The English version is Fisher.
People with the surname A
* Abraham Fischer (1850–1913) South African public official
* A ...
.) She showed this was true for ''n'' = 2, 3, or 4. In 1969, R.G. Swan found a counter-example to Noether's problem, with ''n'' = 47 and ''G'' a cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 47 (although this group can be realized as a Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
over the rationals in other ways). The inverse Galois problem remains unsolved.
First epoch (1908–1919): Physics
Noether was brought to Göttingen
Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911.
General information
The ori ...
in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, a geometrical theory of gravitation developed mainly by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, with Noether's first theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether i ...
, which she proved in 1915, but did not publish until 1918. She not only solved the problem for general relativity, but also determined the conserved quantities for ''every'' system of physical laws that possesses some continuous symmetry. Upon receiving her work, Einstein wrote to Hilbert:
For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the system must be conserved.[.] The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the ''physical laws'' governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
of linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
and energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
within this system, respectively.
Noether's theorem has become a fundamental tool of modern theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, both because of the insight it gives into conservation laws, and also, as a practical calculation tool.[ in Teicher (1999).] Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon:If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.
Second epoch (1920–1926): Ascending and descending chain conditions
In this epoch, Noether became famous for her deft use of ascending (''Teilerkettensatz'') or descending (''Vielfachenkettensatz'') chain conditions. A sequence of non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
subsets ''A''1, ''A''2, ''A''3, etc. of a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''S'' is usually said to be ''ascending'', if each is a subset of the next
:
Conversely, a sequence of subsets of ''S'' is called ''descending'' if each contains the next subset:
:
A chain ''becomes constant after a finite number of steps'' if there is an ''n'' such that for all ''m'' ≥ ''n''. A collection of subsets of a given set satisfies the ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects—and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage.
For example: How to use chain conditions to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These conclusions often are crucial steps in a proof.
Many types of objects in abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called ''Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
'' in her honor. By definition, a Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. A Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the proper ...
is a module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
in which every strictly ascending chain of submodules becomes constant after a finite number of steps. A Noetherian space is a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
in which every strictly ascending chain of open subspaces becomes constant after a finite number of steps; this definition makes the spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of a Noetherian ring a Noetherian topological space.
The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and, ''mutatis mutandis
''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used ...
'', the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
over a Noetherian ring.
Another application of such chain conditions is in Noetherian induction
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s&nb ...
—also known as well-founded induction
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s&nbs ...
—which is a generalization of mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that ''S'' is a partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
. One way of proving a statement about the objects of ''S'' is to assume the existence of a counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
and deduce a contradiction, thereby proving the contrapositive
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statem ...
of the original statement. The basic premise of Noetherian induction is that every non-empty subset of ''S'' contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the ''minimal counterexample''. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.
Second epoch (1920–1926): Commutative rings, ideals, and modules
Noether's paper, ''Idealtheorie in Ringbereichen'' (''Theory of Ideals in Ring Domains'', 1921), is the foundation of general commutative ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, and gives one of the first general definitions of a commutative ring. Before her paper, most results in commutative algebra were restricted to special examples of commutative rings, such as polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the ascending chain condition on ideals, every ideal is finitely generated. In 1943, French mathematician Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a foun ...
coined the term, ''Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
'', to describe this property.[.] A major result in Noether's 1921 paper is the Lasker–Noether theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are relate ...
, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings. The Lasker–Noether theorem can be viewed as a generalization of the fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
which states that any positive integer can be expressed as a product of prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s, and that this decomposition is unique.
Noether's work ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'' (''Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields'', 1927) characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
s: integral domains that are Noetherian, 0- or 1- dimensional, and integrally closed in their quotient fields. This paper also contains what now are called the isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist ...
s, which describe some fundamental natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s, and some other basic results on Noetherian and Artinian module
In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if ...
s.
Second epoch (1920–1926): Elimination theory
In 1923–1924, Noether applied her ideal theory to elimination theory
Elimination may refer to:
Science and medicine
*Elimination reaction, an organic reaction in which two functional groups split to form an organic product
*Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about the factorization of polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same do ...
could be carried over directly. Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method of resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
s.
For illustration, a system of equations often can be written in the form M v = 0 where a matrix (or linear transform
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
) M (without the variable ''x'') times a vector v (that only has non-zero powers of ''x'') is equal to the zero vector, 0. Hence, the determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the matrix M must be zero, providing a new equation in which the variable ''x'' has been eliminated.
Second epoch (1920–1926): Invariant theory of finite groups
Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper, Noether found a solution to the finite basis problem for a finite group of transformations ''G'' acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called Noether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to , ''G'', ! (the factorial of the order , ''G'', of the group ''G''). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number , ''G'', , but Noether was not able to determine whether this bound was correct when the characteristic of the field divides , ''G'', ! but not , ''G'', . For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.
In her 1926 paper, Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford conjecture. In this paper Noether also introduced the ''Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negati ...
'', showing that a finitely generated domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
''A'' over a field ''k'' has a set of algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically in ...
elements such that ''A'' is integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
over ''k'' 1, ..., ''x''''n''">'x''1, ..., ''x''''n''
Second epoch (1920–1926): Contributions to topology
As noted by Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
and Hermann Weyl in their obituaries, Noether's contributions to topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
illustrate her generosity with ideas and how her insights could transform entire fields of mathematics. In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be s ...
. An old joke is that "''a topologist cannot distinguish a donut from a coffee mug''", since they can be continuously deformed into one another.
Noether is credited with fundamental ideas that led to the development of algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
from the earlier combinatorial topology In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such ...
, specifically, the idea of homology groups
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
. According to the account of Alexandrov, Noether attended lectures given by Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
Early life and education
Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
and by him in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle" and he continues that,
Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others, and it became a frequent topic of discussion among the mathematicians of Göttingen.[ Hirzebruch, Friedrich. "Emmy Noether and Topology" in .] Noether observed that her idea of a Betti group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolo ...
makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject "bears the imprint of these remarks of Emmy Noether". Noether mentions her own topology ideas only as an aside in a 1926 publication, where she cites it as an application of group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
.
This algebraic approach to topology was also developed independently in Austria
Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
. In a 1926–1927 course given in Vienna
en, Viennese
, iso_code = AT-9
, registration_plate = W
, postal_code_type = Postal code
, postal_code =
, timezone = CET
, utc_offset = +1
, timezone_DST ...
, Leopold Vietoris defined a homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
, which was developed by Walther Mayer
Walther Mayer (11 March 1887 – 10 September 1948) was an Austrian mathematician, born in Graz, Austria-Hungary. With Leopold Vietoris he is the namesake of the Mayer–Vietoris sequence in topology.. He served as an assistant to Albert Einstei ...
, into an axiomatic definition in 1928.
Third epoch (1927–1935): Hypercomplex numbers and representation theory
Much work on hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...
s and group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these results and gave the first general representation theory of groups and algebras.[.]
Briefly, Noether subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
and ideals in rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
satisfying ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
s. This single work by Noether was of fundamental importance for the development of modern algebra.
Third epoch (1927–1935): Noncommutative algebra
Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
, and Helmut Hasse, she founded the theory of central simple algebras.
A paper by Noether, Helmut Hasse, and Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
pertains to division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
s,[.] which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
splits locally everywhere then it splits globally (so is trivial), and from this, deduced their ''Hauptsatz'' ("main theorem"):every finite dimensional central division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
over an algebraic number field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
F splits over a cyclic cyclotomic extension.
These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra ''D'' are splitting fields. This paper also contains the Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras.
The theorem was first published by Thoralf Skolem in 1927 in ...
which states that any two embeddings of an extension of a field ''k'' into a finite-dimensional central simple algebra over ''k'', are conjugate. The Brauer–Noether theorem gives a characterization of the splitting fields of a central division algebra over a field.
Assessment, recognition, and memorials
Noether's work continues to be relevant for the development of theoretical physics and mathematics and she is consistently ranked as one of the greatest mathematicians of the twentieth century. In his obituary, fellow algebraist BL van der Waerden says that her mathematical originality was "absolute beyond comparison", and Hermann Weyl said that Noether "changed the face of algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
by her work". During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians such as Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
, Hermann Weyl, and Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...
.[.]
In a letter to ''The New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid d ...
'', Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
wrote:[. Als]
online
at the MacTutor History of Mathematics archive.
On 2 January 1935, a few months before her death, mathematician Norbert Wiener wrote that
At an exhibition at the 1964 World's Fair
The 1964–1965 New York World's Fair was a world's fair that held over 140 pavilions and 110 restaurants, representing 80 nations (hosted by 37), 24 US states, and over 45 corporations with the goal and the final result of building exhibits or ...
devoted to Modern Mathematicians, Noether was the only woman represented among the notable mathematicians of the modern world.
Noether has been honored in several memorials,
* The Association for Women in Mathematics
The Association for Women in Mathematics (AWM) is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment o ...
holds a Noether Lecture The Noether Lecture is a distinguished lecture series that honors women "who have made fundamental and sustained contributions to the mathematical sciences". The Association for Women in Mathematics (AWM) established the annual lectures in 1980 as t ...
to honor women in mathematics every year; in its 2005 pamphlet for the event, the Association characterizes Noether as "one of the great mathematicians of her time, someone who worked and struggled for what she loved and believed in. Her life and work remain a tremendous inspiration".
* Consistent with her dedication to her students, the University of Siegen
The University of Siegen (german: Universität Siegen) is a public research university located in Siegen, North Rhine-Westphalia and is part of the Deutsche Forschungsgemeinschaft, a society of Germany's leading research universities. The Univers ...
houses its mathematics and physics departments in buildings on ''the Emmy Noether Campus''.
* The German Research Foundation (Deutsche Forschungsgemeinschaft
The German Research Foundation (german: Deutsche Forschungsgemeinschaft ; DFG ) is a German research funding organization, which functions as a self-governing institution for the promotion of science and research in the Federal Republic of Germ ...
) operates the ''Emmy Noether Programme'', providing funding to early-career researchers to rapidly qualify for a leading position in science and research by leading an independent junior research group.
* A street in her hometown, Erlangen, has been named after Emmy Noether and her father, Max Noether.
* The successor to the secondary school she attended in Erlangen has been renamed as ''the Emmy Noether School''.
* A series of high school workshops and competitions are held in her honor in May of each year since 2001, originally hosted by a subsequent woman mathematics Privatdozent of the University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
.
* Perimeter Institute for Theoretical Physics
Perimeter Institute for Theoretical Physics (PI, Perimeter, PITP) is an independent research centre in foundational theoretical physics located in Waterloo, Ontario, Canada. It was founded in 1999. The institute's founding and major benefactor i ...
annually awards Emmy Noether Visiting Fellowships to outstanding female theoretical physicists. Perimeter Institute is also home to the Emmy Noether Council, a group of volunteers made up of international community, corporate and philanthropic leaders work together to increase the number of women in physics and mathematical physics at Perimeter Institute.
* The Emmy Noether Mathematics Institute in Algebra, Geometry and Function Theory in the Department of Mathematics and Computer Science, Bar-Ilan University
Bar-Ilan University (BIU, he, אוניברסיטת בר-אילן, ''Universitat Bar-Ilan'') is a public research university in the Tel Aviv District city of Ramat Gan, Israel. Established in 1955, Bar Ilan is Israel's second-largest academi ...
, Ramat Gan, Israel was jointly founded in 1992 by the university, the German government
The Federal Cabinet or Federal Government (german: link=no, Bundeskabinett or ') is the chief executive body of the Federal Republic of Germany. It consists of the Federal Chancellor and cabinet ministers. The fundamentals of the cabinet's or ...
and the Minerva Foundation
The Minerva Foundation is a US-based Non-profit organization, non-profit, Scientific society, scientific and charitable foundation. It is headquartered in Berkeley, California. It was established in 1983 by Helen and Elwin Marg. Other than the fo ...
with the aim to stimulate research in the above fields and to encourage collaborations with Germany. Its main topics are Algebraic Geometry, Group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and Complex Function Theory
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. Its activities includes local research projects, conferences, short-term visitors, post-doc fellowships, and the Emmy Noether lectures (an annual series of distinguished lectures). ENI is a member of ERCOM: "European Research Centers of Mathematics".
* In 2013, The European Physical Society established the Emmy Noether Distinction for Women in Physics. Winners have included Dr Catalina Curceanu, Prof Sibylle Günter
Sibylle Günter (born 20 April 1964) is a German theoretical physicist researching tokamak plasmas. Since February 2011, she has headed the Max Planck Institute for Plasma Physics. In October 2015, she was elected a member of the Academia Europa ...
and Prof Anne L'Huillier
Anne L'Huillier (born 1958 in Paris) is a French physicist, and professor of atomic physics at Lund University in Sweden.
Life
L'Huillier first pursued an education and was awarded a Master of Science in theoretical physics and mathematics, bu ...
.
In fiction, Emmy Nutter, the physics professor in "The God Patent" by Ransom Stephens
Ransom Stephens is an American scientist and author.
Professional life
As a particle physicist, Ransom Stephens worked on experiments at SLAC, Fermilab ( DØ), CERN (ATLAS), and Cornell (CLEO), discovered a new type of matter, and worked on the ...
, is based on Emmy Noether.
Farther from home,
* The crater Nöther on the far side of the Moon
The far side of the Moon is the lunar hemisphere that always faces away from Earth, opposite to the near side, because of synchronous rotation in the Moon's orbit. Compared to the near side, the far side's terrain is rugged, with a multitu ...
is named after her.
* The minor planet 7001 Noether is named for Emmy Noether.
* Google
Google LLC () is an American Multinational corporation, multinational technology company focusing on Search Engine, search engine technology, online advertising, cloud computing, software, computer software, quantum computing, e-commerce, ar ...
put a memorial doodle
A doodle is a drawing made while a person's attention is otherwise occupied. Doodles are simple drawings that can have concrete representational meaning or may just be composed of random and abstract lines or shapes, generally without ever lift ...
created by Google artist Sophie Diao on Google's homepage in many countries on 23 March 2015 to celebrate Emmy Noether's 133rd birthday.Google Doodles: Emmy Noether's 133rd Birthday
23 March 2015.
* On 6 November 2020, a satellite named after her (
ÑuSat 13 or "Emmy", COSPAR 2020-079E) was launched into space.
List of doctoral students
Eponymous mathematical topics
*
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
*
Noetherian group
*
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
*
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the proper ...
*
Noetherian space
*
Noetherian induction
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s&nb ...
*
Noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thu ...
*
Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negati ...
*
Noether problem
*
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
*
Noether's second theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations.
:Translated in The action ''S'' of a physical system is an integral of a so-called Lagrangian f ...
*
Lasker–Noether theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are relate ...
*
Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras.
The theorem was first published by Thoralf Skolem in 1927 in ...
*
Brauer–Noether theorem
*
Albert–Brauer–Hasse–Noether theorem
See also
*
Timeline of women in science
This is a timeline of women in science, spanning from ancient history up to the 21st century. While the timeline primarily focuses on women involved with natural sciences such as astronomy, biology, chemistry and physics, it also includes women f ...
Notes
References
Selected works by Emmy Noether (in German)
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* Original German image with link to Tavel's English translation
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Additional sources
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* . Reprinted in
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* Reprinted as an appendix in .
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External links
;Personal documents
* . Noether's application for admission to the
University of Erlangen
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
and three
curricula vitae, two of which are shown in handwriting, with transcriptions. The first of these is in Emmy Noether's own handwriting.
*
;Photographs
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;Academic biographies
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* .
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2019 Interdisciplinary conference on the occasion of the 100th anniversary of Emmy Noether's habilitation organized by
:de:Exzellenzcluster MATH+; Central Women's Representative,
:de:Freie Universität Berlin and
:de:Max-Planck-Institut für Wissenschaftsgeschichte (in German)
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;Newspaper articles
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;Audio discussions
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{{DEFAULTSORT:Noether, Emmy
1882 births
1935 deaths
20th-century German inventors
20th-century German mathematicians
20th-century German scientists
20th-century German physicists
20th-century women scientists
20th-century women mathematicians
20th-century German women
Converts to Lutheranism from Judaism
Algebraists
Bryn Mawr College faculty
Institute for Advanced Study visiting scholars
Jewish emigrants from Nazi Germany to the United States
German Jews
German women mathematicians
German women physicists
Women inventors
Jewish women scientists
Jewish physicists
Jewish scientists
People from Erlangen
People from the Kingdom of Bavaria
University of Göttingen faculty
University of Erlangen-Nuremberg alumni