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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 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 ...
, discoverer of Abelian functions. 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, ...
. 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. ...
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 name ...
, 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 Pri ...
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, Norway, as the second child of the pastor Søren Georg Abel 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 ad ...
. 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 o ...
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 th ...
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. ...
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 ad ...
. 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 pari ...
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 ...
. Meetings of the Storting were held until 1866 in the main hall of the Cathedral School 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, 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, κατηχέω, "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 adul ...
from 1806. The argument was well covered in the press. Søren was given the nickname "Abel Treating" ''( Norwegian: "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 t ...
; 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. 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 ...
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 ...
, found no errors in Abel's formulas, and sent the work on to the leading mathematician in the Nordic countries, Carl Ferdinand Degen 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 ...
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
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: 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 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 a ...
and visit Ferdinand Degen and other mathematicians there. While in Copenhagen, Abel did some work on
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
. 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 t ...
, 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 curren ...
, 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 c ...
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 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 ...
in
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 ...
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 educat ...
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 kn ...
, who was then about to publish his mathematical journal, ''
Journal für die reine und angewandte Mathematik ''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 ...
''. 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 ...
and
Freiberg Freiberg is a university and former mining town in Saxony, Germany. It is a so-called ''Große Kreisstadt'' (large county town) and the administrative centre of Mittelsachsen district. Its historic town centre has been placed under heritage c ...
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 ...
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 functions. 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 at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...
, 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. H ...
. 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 Norges Bank / Noregs Bank is the central bank of Norway. The bank shall promote economic stability in Norway. Norges Bank also manages the Government Pension Fund of Norway and the bank’s own foreign exchange reserves. History The history of ...
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 journ ...
'' 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, ...
. At Christmas 1828, he traveled by sled to
Froland Froland is a List of municipalities of Norway, municipality in Agder counties of Norway, county, Norway. It is part of the Districts of Norway, traditional region of Southern Norway, Sørlandet. The administrative center is the village of Blakst ...
, 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 ...
. 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 Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
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 ...
, or any general
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
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 groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, 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 ...
as well. Abel sent a paper on the unsolvability of the quintic equation to
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 ...
, 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 ...
's result which had held only for
rationals 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 rationa ...
. Abel wrote a fundamental work on the theory of elliptic integrals, 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 ...
. 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 name ...
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. H ...
as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). 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 functions. In 1823 Abel wrote a paper titled "a general representation of the possibility to integrate all differential formulas" ( Norwegian: ''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 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 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 (3 ...
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 ministe ...
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 S ...
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 comm ...
,
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
, 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 functi ...
). 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 Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd ...
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 Pri ...
was established in his memory. Mathematician
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 ...
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 produc ...
. File:Abelmonumentet av Vigeland. - no-nb digifoto 20150625 00154 NB MIT FNR 11682.jpg, Abelmonumentet in front of the
Royal Palace, Oslo The Royal Palace ( no, Slottet or ''Det kongelige slott'') in Oslo was built in the first half of the 19th century as the Norwegian residence of the French-born King Charles III John, who reigned as king of Norway and Sweden. The palace is the ...
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 vill ...
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/ Osedalen and about southeast of the village of M ...
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

* List of things named after Niels Henrik Abel *
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 radical ...


Notes


References


Further reading

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External links


Biographies and handwritten manuscripts
from the Abel Prize website *

* *
Translation of Niels Henrik Abel's "Research on Elliptic Functions"
a
Convergence

Famous Quotes by Niels Henrik Abel
a
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