Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian
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 pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the
general 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 q ...
in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of
elliptic functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
, discoverer of
Abelian function
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s. He made his discoveries while living in poverty and died at the age of 26 from
tuberculosis
Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in ...
.
Most of his work was done in six or seven years of his working life.
Regarding Abel, the French mathematician
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
said: "Abel has left mathematicians enough to keep them busy for five hundred years."
Another French mathematician,
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named ...
, said: "What a head the young Norwegian has!"
The
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
in mathematics, originally proposed in 1899 to complement the
Nobel Prize
The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
s (but first awarded in 2003), is named in his honour.
Life
Early life
Niels Henrik Abel was born in
Nedstrand
Nedstrand (locally, ''Stranda'') is a village in Tysvær municipality in Rogaland county, Norway. The village is on the Nedstrand peninsula's southeast coast, at the confluence of the Nedstrandsfjorden and Vindafjorden. The village of Hindarå ...
, Norway, as the second child of the pastor
Søren Georg Abel
Søren Georg Abel (3 January 1772 – 5 May 1820) was a Norwegian priest and politician, also known as the father of mathematician Niels Henrik Abel.
Personal life
Søren Georg Abel was born on 3 January 1772 in Mo, Telemark as the son of Hans Ma ...
and Anne Marie Simonsen. When Niels Henrik Abel was born, the family was living at a rectory on
Finnøy
Finnøy is a former municipality in Rogaland county, Norway. The island municipality existed from 1838 until its dissolution in 2020 when it was merged into Stavanger Municipality. It was located in the traditional district of Ryfylke. The a ...
. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the
bailiff
A bailiff (from Middle English baillif, Old French ''baillis'', ''bail'' "custody") is a manager, overseer or custodian – a legal officer to whom some degree of authority or jurisdiction is given. Bailiffs are of various kinds and their offi ...
in Nedstrand in July / August of his year of birth.
[
Stubhaug, Arild]
Niels Henrik Abel – utdypning (NBL-artikkel)
– Store norske leksion
Niels Henrik Abel's father, Søren Georg Abel, had a degree in
theology
Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...
and
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
and served as pastor at
Finnøy
Finnøy is a former municipality in Rogaland county, Norway. The island municipality existed from 1838 until its dissolution in 2020 when it was merged into Stavanger Municipality. It was located in the traditional district of Ryfylke. The a ...
. Søren's father, Niels's grandfather, Hans Mathias Abel, was also a pastor, at
Gjerstad Church
Gjerstad Church ( no, Gjerstad kirke, locally: ''Gjerstad kjørke'') is a parish church of the Church of Norway in Gjerstad Municipality in Agder county, Norway. It is located in the village of Gjerstad. It is the church for the Gjerstad parish ...
near the
town of Risør. Søren had spent his childhood at Gjerstad, and had also served as chaplain there; and after his father's death in 1804, Søren was appointed pastor at Gjerstad and the family moved there. The Abel family originated in
Schleswig
The Duchy of Schleswig ( da, Hertugdømmet Slesvig; german: Herzogtum Schleswig; nds, Hartogdom Sleswig; frr, Härtochduum Slaswik) was a duchy in Southern Jutland () covering the area between about 60 km (35 miles) north and 70 km ...
and came to Norway in the 17th century.
Anne Marie Simonsen was from
Risør
is a municipality in Agder county, Norway. It is located on the Skagerrak coast in the traditional region of Sørlandet. The administrative centre of Risør municipality is the town of Risør. There are many villages in Risør such as Akland, ...
; her father, Niels Henrik Saxild Simonsen, was a tradesman and merchant ship-owner, and said to be the richest person in Risør. Anne Marie had grown up with two
stepmothers, in relatively luxurious surroundings. At Gjerstad rectory, she enjoyed arranging balls and social gatherings. Much suggests she was early on an alcoholic and took little interest in the upbringing of the children. Niels Henrik and his brothers were given their schooling by their father, with handwritten books to read. An addition table in a book of mathematics reads: 1+0=0.
Cathedral School and Royal Frederick University
With Norwegian independence and the first election held in Norway, in 1814, Søren Abel was elected as a representative to the
Storting
The Storting ( no, Stortinget ) (lit. the Great Thing) is the supreme legislature of Norway, established in 1814 by the Constitution of Norway. It is located in Oslo. The unicameral parliament has 169 members and is elected every four years bas ...
. Meetings of the Storting were held until 1866 in the main hall of the
Cathedral School
Cathedral schools began in the Early Middle Ages as centers of advanced education, some of them ultimately evolving into medieval universities. Throughout the Middle Ages and beyond, they were complemented by the monastic schools. Some of these e ...
in Christiania (now known as
Oslo
Oslo ( , , or ; sma, Oslove) is the capital and most populous city of Norway. It constitutes both a county and a municipality. The municipality of Oslo had a population of in 2022, while the city's greater urban area had a population of ...
). Almost certainly, this is how he came into contact with the school, and he decided that his eldest son, Hans Mathias, should start there the following year. However, when the time for his departure approached, Hans was so saddened and depressed over having to leave home that his father did not dare send him away. He decided to send Niels instead.
In 1815, Niels Abel entered the Cathedral School at the age of 13. His elder brother Hans joined him there a year later. They shared rooms and had classes together. Hans got better grades than Niels; however, a new mathematics teacher,
Bernt Michael Holmboe
Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810. Following a short period at the Royal Frederick University, which included a ...
, was appointed in 1818. He gave the students mathematical tasks to do at home. He saw Niels Henrik's talent in mathematics, and encouraged him to study the subject to an advanced level. He even gave Niels private lessons after school.
In 1818, Søren Abel had a public theological argument with the theologian Stener Johannes Stenersen regarding his
catechism
A catechism (; from grc, κατηχÎω, "to teach orally") is a summary or exposition of doctrine and serves as a learning introduction to the Sacraments traditionally used in catechesis, or Christian religious teaching of children and adult c ...
from 1806. The argument was well covered in the press. Søren was given the nickname "Abel Treating" ''(
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
*Demographics of Norway
*The Norwegian language, including the ...
: "Abel Spandabel")''. Niels' reaction to the quarrel was said to have been "excessive gaiety". At the same time, Søren also almost faced impeachment after insulting
Carsten Anker
Carsten Tank Anker (17 November 1747 – 13 March 1824) was a Norwegian businessman, civil servant, politician and one of the Fathers of the Constitution of Norway. He was the owner of the manor house in Akershus at which the original National As ...
, the host of the
Norwegian Constituent Assembly
The Norwegian Constituent Assembly (in Norwegian ''Grunnlovsforsamlingen'', also known as ''Riksforsamlingen'') is the name given to the 1814 constitutional assembly at Eidsvoll in Norway, that adopted the Norwegian Constitution and formalised th ...
; and in September 1818 he returned to Gjerstad with his political career in ruins. He began drinking heavily and died only two years later, in 1820, aged 48.
Bernt Michael Holmboe supported Niels Henrik Abel with a scholarship to remain at the school and raised money from his friends to enable him to study at the
Royal Frederick University
The University of Oslo ( no, Universitetet i Oslo; la, Universitas Osloensis) is a public research university located in Oslo, Norway. It is the highest ranked and oldest university in Norway. It is consistently ranked among the top universit ...
.
When Abel entered the university in 1821, he was already the most knowledgeable mathematician in Norway. Holmboe had nothing more he could teach him and Abel had studied all the latest mathematical literature in the university library. During that time, Abel started working on 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 q ...
in radicals. Mathematicians had been looking for a solution to this problem for over 250 years. In 1821, Abel thought he had found the solution. The two professors of mathematics in Christiania, Søren Rasmussen and
Christopher Hansteen
Christopher Hansteen (26 September 1784 – 11 April 1873) was a Norwegian geophysicist, astronomer and physicist, best known for his mapping of Earth's magnetic field.
Early life and career
Hansteen was born in Christiania as the son of J ...
, found no errors in Abel's formulas, and sent the work on to the leading mathematician in the Nordic countries,
Carl Ferdinand Degen
Carl Ferdinand Degen (1 November 1766 – 8 April 1825) was a Danish mathematician. His most important contributions were within number theory and he advised the young, aspiring Norwegian mathematician Niels Henrik Abel in a decisive way. Degen has ...
in Copenhagen. He too found no faults but still doubted that the solution, which so many outstanding mathematicians had sought for so long, could really have been found by an unknown student in far-off Christiania. Degen noted, however, Abel's unusually sharp mind, and believed that such a talented young man should not waste his abilities on such a "sterile object" as the fifth degree equation, but rather on
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s and
transcendence
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
; for then, wrote Degen, he would "discover Magellanian thoroughfares to large portions of a vast analytical ocean".
Degen asked Abel to give a numerical example of his method. While trying to provide an example, Abel found a mistake in his paper.
[Abel biography](_blank)
Groups.dcs.st-and.ac.uk. Retrieved on 2011-07-12. This led to a discovery in 1823 that a solution to a fifth- or higher-degree equation was impossible.
Abel graduated in 1822. His performance was exceptionally high in mathematics and average in other matters.
Career
After he graduated, professors from university supported Abel financially, and Professor Christopher Hansteen let him live in a room in the attic of his home. Abel would later view Ms. Hansteen as his second mother. While living here, Abel helped his younger brother, Peder Abel, through
examen artium
Examen artium was the name of the academic certification conferred in Denmark and Norway, qualifying the student for admission to university studies. Examen artium was originally introduced as the entrance exam of the University of Copenhagen in 1 ...
. He also helped his sister Elisabeth to find work in the town.
In early 1823, Niels Abel published his first article in "Magazin for Naturvidenskaberne", Norway's first scientific journal, which had been co-founded by Professor Hansteen. Abel published several articles, but the journal soon realized that this was not material for the common reader. In 1823, Abel also wrote a paper in French. It was "a general representation of the possibility to integrate all differential formulas" (''
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
*Demographics of Norway
*The Norwegian language, including the ...
: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler)''. He applied for funds at the university to publish it. However, the work was lost while being reviewed, never to be found thereafter.
In mid-1823, Professor Rasmussen gave Abel a gift of 100
speciedaler
The rigsdaler specie was a unit of silver currency used in Norway, renamed as the speciedaler in 1816 and used until 1873. Norway used a common reichsthaler currency system shared with Denmark, Hamburg and Schleswig-Holstein until 1873 when the go ...
so he could travel to
Copenhagen
Copenhagen ( or .; da, København ) is the capital and most populous city of Denmark, with a proper population of around 815.000 in the last quarter of 2022; and some 1.370,000 in the urban area; and the wider Copenhagen metropolitan ar ...
and visit Ferdinand Degen and other mathematicians there. While in Copenhagen, Abel did some work on
Fermat's Last Theorem. Abel's uncle, Peder Mandrup Tuxen, lived at the naval base in
Christianshavn
Christianshavn (literally, "ingChristian's Harbour") is a neighbourhood in Copenhagen, Denmark. Part of the Indre By District, it is located on several artificial islands between the islands of Zealand and Amager and separated from the rest of th ...
, Copenhagen, and at a ball there Niels Abel met Christine Kemp, his future fiancée. In 1824, Christine moved to
Son
A son is a male offspring; a boy or a man in relation to his parents. The female counterpart is a daughter. From a biological perspective, a son constitutes a first degree relative.
Social issues
In pre-industrial societies and some current c ...
, Norway to work as a governess and the couple got engaged over Christmas.
After returning from Copenhagen, Abel applied for a government scholarship in order to visit top mathematicians in Germany and France, but he was instead granted 200 speciedaler yearly for two years, to stay in Christiania and study German and French. In the next two years, he was promised a scholarship of 600 speciedaler yearly and he would then be permitted to travel abroad.
While studying these languages, Abel published his first notable work in 1824, ''Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré'' (Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven). By 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals (now referred to as the
Abel–Ruffini theorem). However, this paper was in an abstruse and difficult form, in part because he had restricted himself to only six pages in order to save money on printing. A more detailed
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
was published in 1826 in the first volume of ''
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
''.
In 1825, Abel wrote a personal letter to
King Carl Johan of Norway/Sweden requesting permission to travel abroad. He was granted this permission, and in September 1825 he left Christiania together with four friends from university (Christian P.B Boeck, Balthazar M. Keilhau, Nicolay B. Møller and Otto Tank). These four friends of Abel were traveling to Berlin and to the Alps to study geology. Abel wanted to follow them to Copenhagen and from there make his way to Göttingen. The terms for his scholarship stipulated that he was to visit
Gauss in
Göttingen
Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
and then continue to Paris. However, when he got as far as Copenhagen, he changed his plans. He wanted to follow his friends to Berlin instead, intending to visit Göttingen and Paris afterwards.
On the way, he visited the astronomer
Heinrich Christian Schumacher
Prof Heinrich Christian Schumacher FRS(For) FRSE (3 September 1780 – 28 December 1850) was a German-Danish astronomer and mathematician.
Biography
Schumacher was born at Bramstedt, in Holstein, near the German/Danish border. He was educated ...
in Altona, now a district of Hamburg. He then spent four months in Berlin, where he became well acquainted with
August Leopold Crelle
August Leopold Crelle (17 March 1780 – 6 October 1855) was a German mathematician. He was born in Eichwerder near Wriezen, Brandenburg, and died in Berlin. He is the founder of ''Journal für die reine und angewandte Mathematik'' (also know ...
, who was then about to publish his mathematical journal, ''
Journal für die reine und angewandte Mathematik''. This project was warmly encouraged by Abel, who contributed much to the success of the venture. Abel contributed seven articles to it in its first year.
From Berlin Abel also followed his friends to the Alps. He went to
Leipzig
Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as wel ...
and
Freiberg to visit
Georg Amadeus Carl Friedrich Naumann
Georg Amadeus Carl Friedrich Naumann (30 May 1797 – 26 November 1873), also known as Karl Friedrich Naumann, was a German mineralogist and geologist. The crater Naumann on the Moon is named after him.
Life
Naumann was born at Dresden, the s ...
and his brother the mathematician August Naumann. In Freiberg Abel did research in the theory of functions, particularly,
elliptic,
hyperelliptic, and a new class now known as
abelian function
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s.
From Freiberg they went on to Dresden, Prague, Vienna, Trieste, Venice, Verona, Bolzano, Innsbruck, Luzern and Basel. From July 1826 Abel traveled on his own from Basel to Paris. Abel had sent most of his work to Berlin to be published in Crelle's Journal, but he had saved what he regarded as his most important work for the
French Academy of Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...
, a theorem on addition of algebraic differentials. With the help of a painter,
Johan Gørbitz, he found an apartment in Paris and continued his work on the theorem. He finished in October 1826 and submitted it to the academy. It was to be reviewed by
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
. Abel's work was scarcely known in Paris, and his modesty restrained him from proclaiming his research. The theorem was put aside and forgotten until his death.
Abel's limited finances finally compelled him to abandon his tour in January 1827. He returned to Berlin, and was offered a position as editor of Crelle's Journal, but opted out. By May 1827 he was back in Norway. His tour abroad was viewed as a failure. He had not visited Gauss in Göttingen and he had not published anything in Paris. His scholarship was therefore not renewed and he had to take up a private loan in
Norges Bank of 200 spesidaler. He never repaid this loan. He also started tutoring. He continued to send most of his work to Crelle's Journal. But in mid-1828 he published, in rivalry with
Carl Jacobi, an important work on elliptic functions in ''
Astronomische Nachrichten
''Astronomische Nachrichten'' (''Astronomical Notes''), one of the first international journals in the field of astronomy, was established in 1821 by the German astronomer Heinrich Christian Schumacher. It claims to be the oldest astronomical jour ...
'' in Altona.
Death
While in Paris, Abel contracted
tuberculosis
Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, in ...
. At Christmas 1828, he traveled by sled to
Froland, Norway, to visit his fiancée. He became seriously ill on the journey. Although a temporary improvement allowed the couple to enjoy the holiday together, he died relatively soon after on 6 April 1829, just two days before a letter arrived from August Crelle. Crelle had been searching for a new job for Abel in Berlin and had actually managed to have him appointed as a Professor at the
University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
. Crelle wrote to Abel to tell him, but the good news came too late.
Contributions to mathematics
Abel showed that there is no general
algebraic solution for the roots of a
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 q ...
, or any general
polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of
Galois) a branch of mathematics known as
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 ...
, which is invaluable not only in many areas of mathematics, but for much of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
as well. Abel sent a paper on the unsolvability of the quintic equation to
Carl Friedrich Gauss, who proceeded to discard without a glance what he believed to be the worthless work of a crank.
As a 16-year-old, Abel gave a rigorous proof of the
binomial theorem valid for all numbers, extending
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
's result which had held only for
rationals. Abel wrote a fundamental work on the theory of
elliptic integrals
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
, containing the foundations of the theory of
elliptic functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
.
While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named ...
later described to
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet
Horatius Horatius may refer to:
People Roman era
* several ancient Roman men of the '' gens Horatia'', including:
** Quintus Horatius Flaccus, the poet known in English as Horace
** one of the Horatii, three members of the ''gens Horatia'' who fought to the ...
). The paper was, however, misplaced by Cauchy.
While abroad Abel had sent most of his work to Berlin to be published in the ''Crelle's Journal'', but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. The theorem was put aside and forgotten until his death. While in Freiberg, Abel did research in the theory of functions, particularly,
elliptic,
hyperelliptic, and a new class now known as
abelian function
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s.
In 1823 Abel wrote a paper titled "a general representation of the possibility to integrate all differential formulas" (
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
*Demographics of Norway
*The Norwegian language, including the ...
: ''en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler''). He applied for funds at the university to publish it. However the work was lost, while being reviewed, never to be found thereafter.
Abel said famously of
Carl Friedrich Gauss's writing style, "He is like the fox, who effaces his tracks in the sand with his tail." Gauss replied to him by saying, "No self-respecting architect leaves the scaffolding in place after completing his building."
Legacy
Under Abel's guidance, the prevailing obscurities of
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 (38 ...
began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made. His works, the greater part of which originally appeared in ''
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
'', were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by
Ludwig Sylow
Peter Ludwig Mejdell Sylow () (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory. Biography
He was born and died in Christiania (now Oslo). Sylow was a son of government minister ...
and
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Life and career
Marius Sophu ...
was published in 1881. The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (e.g.,
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
,
abelian category, and
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
).
On 6 April 1929, four Norwegian stamps were issued for the centenary of Abel's death. His portrait appears on the 500-
kroner banknote
A banknote—also called a bill (North American English), paper money, or simply a note—is a type of negotiable instrument, negotiable promissory note, made by a bank or other licensed authority, payable to the bearer on demand.
Banknotes w ...
(version V) issued during 1978–1985. On 5 June 2002, four Norwegian stamps were issued in honour of Abel two months before the bicentenary of his birth. There is also a 20-kroner coin issued by Norway in his honour. A statue of Abel stands in Oslo, and crater
Abel on the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
was named after him. In 2002, the
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
was established in his memory.
Mathematician
Felix Klein wrote about Abel:
File:NOK 500 V recto.jpg, Niels Henrik Abel on a Norwegian 500 kroner banknote, 1978
File:Abelmonumentet 1.jpg, Statue of Niels Henrik Abel, called the "Abelmonumentet", in Oslo (former Christiania) by Gustav Vigeland
Gustav Vigeland (11 April 1869 – 12 March 1943), born as Adolf Gustav Thorsen, was a Norwegian sculptor. Gustav Vigeland occupies a special position among Norwegian sculptors, both in the power of his creative imagination and in his product ...
.
File:Abelmonumentet av Vigeland. - no-nb digifoto 20150625 00154 NB MIT FNR 11682.jpg, Abelmonumentet in front of the Royal Palace, Oslo
File:Holmengard-119.JPG, Niels Henrik Abel memorial in Gjerstad
Gjerstad is a municipality in Agder county, Norway. It is located in the traditional region of Sørlandet, in the southeastern part of the county, along the border with Telemark county. The administrative centre of the municipality is the villa ...
File:Abelmonument.jpg, A monument for Abel at Frolands verk
Frolands verk is a small village area in Froland municipality in Agder county, Norway. The village is located along the Norwegian County Road 42, almost west of the municipal centre of Blakstad, Agder, Blakstad/Osedalen and about southeast of th ...
by Gustav Lærum
File:Stamps of Norway, 1929-Niels Henrik Abel1.jpg
File:Stamps of Norway, 1929-Niels Henrik Abel2.jpg
File:Stamps of Norway, 1929-Niels Henrik Abel3.jpg
File:Stamps of Norway, 1929-Niels Henrik Abel4.jpg
See also
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List of things named after Niels Henrik Abel
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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, ...
Notes
References
Further reading
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External links
Biographies and handwritten manuscripts from the Abel Prize website
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Translation of Niels Henrik Abel's "Research on Elliptic Functions"a
ConvergenceFamous Quotes by Niels Henrik Abela
Convergence
The Niels Henrik Abel mathematical contest The Norwegian Mathematical Olympiad
Family genealogy*
{{DEFAULTSORT:Abel, Niels Henrik
1802 births
1829 deaths
19th-century deaths from tuberculosis
19th-century Norwegian mathematicians
Group theorists
Norwegian Lutherans
Norwegian mathematicians
People educated at Oslo Cathedral School
People from Froland
Tuberculosis deaths in Norway
University of Oslo alumni