Leopold Kronecker
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Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German
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, mathematical structure, structure, space, Mathematica ...
who worked on
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
,
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
. He criticized
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
's work on
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, and was quoted by as having said, "'" ("God made the integers, all else is the work of man").The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886".

Weber, Heinrich L. 1891–1892
Kronecker''Jahresbericht der Deutschen Mathematiker-Vereinigung''
2:5-23. (The quote is on p. 19.)
Kronecker was a student and lifelong friend of
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
.


Biography

Leopold Kronecker was born on 7 December 1823 in
Liegnitz Legnica (Polish: ; german: Liegnitz, szl, Lignica, cz, Lehnice, la, Lignitium) is a city in southwestern Poland, in the central part of Lower Silesia, on the Kaczawa River (left tributary of the Oder) and the Czarna Woda. Between 1 June 19 ...
,
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 ...
(now Legnica,
Poland Poland, officially the Republic of Poland, , is a country in Central Europe. Poland is divided into Voivodeships of Poland, sixteen voivodeships and is the fifth most populous member state of the European Union (EU), with over 38 mill ...
) in a wealthy
Jewish Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
family. His parents, Isidor and Johanna (née Prausnitzep), took care of their children's education and provided them with private tutoring at home—Leopold's younger brother Hugo Kronecker would also follow a scientific path, later becoming a notable
physiologist Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
. Kronecker then went to the Liegnitz Gymnasium where he was interested in a wide range of topics including
science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...
,
history History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the History of writing#Inventions of writing, invention of writing systems is considered prehistory. "History" is an umbr ...
and philosophy, while also practicing gymnastics and swimming. At the gymnasium he was taught by
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...
, who noticed and encouraged the boy's interest in mathematics. In 1841 Kronecker became a student at the
University of Berlin The Humboldt University of Berlin (german: link=no, Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick Will ...
where his interest did not immediately focus on mathematics, but rather spread over several subjects including
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and philosophy. He spent the summer of 1843 at the
University of Bonn The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine ...
studying astronomy and 1843–44 at the University of Breslau following his former teacher Kummer. Back in Berlin, Kronecker studied mathematics with Peter Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlet's supervision. After obtaining his degree, Kronecker did not follow his interest in research on an academic career path. He went back to his hometown to manage a large farming estate built up by his mother's uncle, a former banker. In 1848 he married his cousin Fanny Prausnitzer, and the couple had six children. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results. In 1853 he wrote a memoir on the algebraic solvability of equations extending the work of
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, ...
on the theory of equations. Due to his business activity, Kronecker was financially comfortable, and thus he could return to Berlin in 1855 to pursue mathematics as a private scholar. Dirichlet, whose wife Rebecka came from the wealthy Mendelssohn family, had introduced Kronecker to the Berlin elite. He became a close friend of
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematic ...
, who had recently joined the university, and his former teacher Kummer who had just taken over Dirichlet's mathematics chair. Over the following years Kronecker published numerous papers resulting from his previous years' independent research. As a result of this published research, he was elected a member of the Berlin Academy in 1861. Although he held no official university position, Kronecker had the right as a member of the Academy to hold classes at the University of Berlin and he decided to do so, starting in 1862. In 1866, when
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first r ...
died, Kronecker was offered the mathematics chair 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 i ...
(previously held by
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 refe ...
and Dirichlet), but he refused, preferring to keep his position at the Academy. Only in 1883, when Kummer retired from the university, was Kronecker invited to succeed him and became an ordinary professor. Kronecker was the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, and Franz Mertens, amongst others. His philosophical view of mathematics put him in conflict with several mathematicians over the years, notably straining his relationship with Weierstrass, who almost decided to leave the university in 1888. Kronecker died on 29 December 1891 in Berlin, several months after the death of his wife. In the last year of his life, he converted to
Christianity Christianity is an Abrahamic monotheistic religion based on the life and teachings of Jesus of Nazareth Jesus, likely from he, יֵשׁוּעַ, translit=Yēšūaʿ, label=Hebrew/Aramaic ( AD 30 or 33), also referred to as Jesu ...
. He is buried in the Alter St Matthäus Kirchhof cemetery in
Berlin Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...
-
Schöneberg Schöneberg () is a locality of Berlin, Germany. Until Berlin's 2001 administrative reform it was a separate borough including the locality of Friedenau. Together with the former borough of Tempelhof it is now part of the new borough of Te ...
, close to
Gustav Kirchhoff Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coin ...
.


Scientific activity


Mathematics research

An important part of Kronecker's research focused on
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
and
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
. In an 1853 paper on the theory of equations and
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 t ...
he formulated the Kronecker–Weber theorem, without however offering a definitive proof (the theorem was proved completely much later by David Hilbert). He also introduced the structure theorem for finitely-generated abelian groups. Kronecker studied elliptic functions and conjectured his " liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem. In an 1850 paper, ''On the Solution of the General Equation of the Fifth Degree'', Kronecker solved the
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 ...
by applying
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 ...
(though his solution was not in terms of radicals: that was already proven impossible by the
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means ...
). In algebraic number theory Kronecker introduced the theory of divisors as an alternative to Dedekind's theory of
ideals Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
, which he did not find acceptable for philosophical reasons. Although the general adoption of Dedekind's approach led Kronecker's theory to be ignored for a long time, his divisors were found useful and were revived by several mathematicians in the 20th century. Kronecker also contributed to the concept of continuity, reconstructing the form of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s in
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. In
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematic ...
. Also named for Kronecker are the Kronecker limit formula,
Kronecker's congruence In mathematics, Kronecker's congruence, introduced by Kronecker, states that : \Phi_p(x,y)\equiv (x-y^p)(x^p-y)\bmod p, where ''p'' is a prime and Φ''p''(''x'',''y'') is the modular polynomial of order ''p'', given by :\Phi_n(x,j) = \prod_\tau ( ...
,
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 ...
,
Kronecker comb In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\t ...
, Kronecker symbol, Kronecker product,
Kronecker's method 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 doma ...
for factorizing polynomials, Kronecker substitution, Kronecker's theorem in number theory, Kronecker's lemma, and
Eisenstein–Kronecker number In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers. They are defined in terms of classical Eisenstein–Kronecker series, which were studied by Kenichi Bannai and Shinich ...
s.


Philosophy of mathematics

Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics.


Honours

Kronecker was elected as a member of several academies: *
Prussian Academy of Sciences The Royal Prussian Academy of Sciences (german: Königlich-Preußische Akademie der Wissenschaften) was an academy established in Berlin, Germany on 11 July 1700, four years after the Prussian Academy of Arts, or "Arts Academy," to which "Berlin ...
(1861) *
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at th ...
(1868) *
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, r ...
(1884). The
25624 Kronecker 56 may refer to: * 56 (number) * one of the years 56 BC, AD 56, 1856, 1956, 2056 * 56.com, a Chinese online video platform * Fiftysix, Arkansas, unincorporated community in United States * Fifty-Six, Arkansas, city in United States * "Fifty Six", ...
asteroid is named after him.


Publications

* *


References


Further reading

* * * * * * *


External links

* * {{DEFAULTSORT:Kronecker, Leopold 1823 births 1891 deaths 19th-century German mathematicians Converts to Christianity Corresponding members of the Saint Petersburg Academy of Sciences Foreign Members of the Royal Society 19th-century German Jews Humboldt University of Berlin alumni Humboldt University of Berlin faculty Independent scientists Number theorists Intuitionism People from Legnica People from the Province of Silesia