**Niels Henrik Abel** was a Norwegian

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

who proved the impossibility of solving the

quintic equationIn mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

in

radicalsIn mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

.

### Early life

Niels Henrik Abel was born in

NedstrandNedstrand is a village and a former municipality in Rogaland county, Norway.The village is located on the south-eastern coast of Nedstrandhalvøya, and its population is 241 ....

, Norway, as second child to

Søren Georg AbelSøren Georg Abel was a Norwegian priest and politician, also known as the father of mathematician Niels Henrik Abel.-Personal life:...

and Anne Marie Simonsen. When he was born, the family was living at the rectory at

FinnøyFinnøy is a municipality in Rogaland county, Norway. It is an island community located in Boknafjorden, 13 nautical miles north of Stavanger.The parish of Finnø was established as a municipality on 1 January 1838...

. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the

bailiffA bailiff is a governor or custodian ; a legal officer to whom some degree of authority, care or jurisdiction is committed...

in Nedstrand in July / August of his year of birth.

Peter Schnitler Marstrand was bailiff in Nedstrand from 1798 to 1809. His wife, Madam Anne Margrethe Marstrand, was one of ten sponsors at Abel's baptism. The other sponsors were from Finnøy. Niels Henrik was born premature and, according to oral tradition, a newly born child had to be washed in red wine and wrapped in cotton cloth in order to survive. Niels Henrik's father baptised him, and wrote in the parish register: *"6 September 1802 baptized in Finnøy church the pastor Søren G. Abel and Ane Marie Simonsen's child, Niels Henrik, born 5th August" – (Norwegian: September 6te 1802 døbt i Findøe Kirke Sognepræsten Søren G. Abel og Ane Marie Simonsens Barn Niels Henrik, fød den 5te August."*)

*Source: Skadberg, Gunnar A.: Ætt og heim 2004, chapter 5, Sogneprest Søren Georg Abel – Matematikerens far. ISBN 82-90087-65-9*

Morten Kiærulf was appointed pastor for Nedstrand in 1829. He wrote in a letter in 1880 to professor Bjerknes in Valle, that he was told by locals that one of Abel's sons was born at former bailiff Marstrand's house in Nedstrand.

*Source: Myhre, Sigmar: Ætt og heim 2009, chapter 6, Niels Henrik Abel – fødestaden. ISBN 978-82-90087-72-7*

Niels Henrik Abel's father, Søren Georg Abel, had a degree in

theologyTheology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

and

philosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

and served as pastor at

FinnøyFinnøy is a municipality in Rogaland county, Norway. It is an island community located in Boknafjorden, 13 nautical miles north of Stavanger.The parish of Finnø was established as a municipality on 1 January 1838...

. Søren's father, Niels's grandfather, Hans Mathias Abel, was also a pastor, at

GjerstadGjerstad is a municipality in Aust-Agder county, Norway. It is located in the traditional region of Agder, in the southeastern part of the county, along the border with Telemark county.-General information:...

near

Risøris a city and municipality in Aust-Agder county, Norway. The city belongs to the traditional region of Sørlandet. It is a popular tourist place. The surrounding area includes many small lakes and hills, and is known for its beautiful coastline as well....

. 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.

Anne Marie Simonsen was from Risør; 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

stepmothersA stepfamily, also known as a blended family or reconstituted family, is a family in which one or both members of the couple have children from a previous relationship...

, in relative 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. Interestingly, 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. Meetings of the Storting were held until 1866 in the main hall of the

Cathedral SchoolSchola Osloensis, known in Norwegian as Oslo katedralskole and more commonly as "Katta" is an upper secondary school located in Oslo, Norway. The school offers the college preparatory studiespesialisering of the Norwegian school system...

in Christiania (now known as

OsloOslo is a municipality, as well as the capital and most populous city in Norway. As a municipality , it was established on 1 January 1838. Founded around 1048 by King Harald III of Norway, the city was largely destroyed by fire in 1624. The city was moved under the reign of Denmark–Norway's King...

). 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. In general, Hans got better grades than Niels; however, a new mathematics teacher,

Bernt Michael HolmboeBernt Michael Holmboe was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810...

, 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 Stener Johannes Stenersen regarding his

catechismA catechism , i.e. to indoctrinate) is a summary or exposition of doctrine, traditionally used in Christian religious teaching from New Testament times to the present...

from 1806. The argument was well covered in 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, the host of the Norwegian Constituent Assembly; 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. At his funeral, with the rectory full of guests, the widow Anne Marie Abel got drunk and went openly to bed with one of the servants.

That same year, after receiving news of Søren Georg Abel's death, Peder Mandrup Tuxen wrote to his wife Marie Elisabeth Simonsen (who was Anne Marie Abel's sister):

Often I think of Anne Marie's peculiar character. I have always viewed her as phlegmatic and at least thought she had inferior passions. Well I remember from Finnøy, a handsome peasant, she may have liked and often talked to with great pleasure, but she seemed so cold to me, that I could not view this as nothing other than simple benevolence. Dark in front of me however, I have heard of another such inclination now.

(Norwegian: Ofte tænker jeg paa Anne Maries besyndelige Karakter. Jeg har altid anset hende for flagmatisk og mindst tiltænkt hende saa lave Lidenskaper. Vel mindes jeg fra Findøen, der var en smuk Bondekarl, hun godt kunde lide og hun talte ofte med ham med særdeles Fornøjelse, men hun forekom mig saa kold, at jeg ej kunde tage dette for andet end simpel Velvilje. Dunkelt staar imidlertid for mig, at jeg har hørt noget mere om en saadan Inklination.)

*Source: Skadberg, Gunnar A.: Ætt og heim 2004, chapter 5, Sogneprest Søren Georg Abel – Matematikerens far. ISBN 82-90087-65-9*

The two brothers reacted differently to the decline of their family. At school, Niels Henrik did extremely well in mathematics, though he struggled in other subjects. Hans Mathias, on the other hand, went into a serious depression, never to recover; he had quit school and returned to Gjerstad shortly before their father died. The family was left in strained circumstances. Anne Marie Abel's once-rich father went bankrupt in a recession after the

Napoleonic WarsThe Napoleonic Wars were a series of wars declared against Napoleon's French Empire by opposing coalitions that ran from 1803 to 1815. As a continuation of the wars sparked by the French Revolution of 1789, they revolutionised European armies and played out on an unprecedented scale, mainly due to...

, and died also in 1820.

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 UniversityThe University of Oslo , formerly The Royal Frederick University , is the oldest and largest university in Norway, situated in the Norwegian capital of Oslo. The university was founded in 1811 and was modelled after the recently established University of Berlin...

.

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. Abel had also started work on his first achievement, the

quintic equationIn mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

in radicals. Abel initially thought he had found the solution to the quintic equation in radicals in 1821. Mathematicians had been looking for a solution on this problem for over 250 years. The two professors in Christiania, Søren Rasmussen and

Christopher HansteenChristopher Hansteen was a Norwegian geophysicist, astronomer and physicist, best known for his mapping of Earth's magnetic field.-Early life and career:...

, found no errors in Abel's formulas, and sent the work on to the leading mathematician in the Nordic countries, Professor Ferdinand Degen in Copenhagen. He also found no faults, but still doubted that the solution, which so many outstanding mathematicians had sought for so long, could now 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 functionIn complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s and

transcendenceTranscendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...

; for then, writes Degen, he will "discover Magellanian thoroughfares to large portions of a vast analytical ocean". Degen asked Abel to give a numerical example of his method and, while trying to provide an example, Abel discovered a mistake in his paper.

Abel graduated in 1822. His performance was average, except in mathematics.

### 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 to

examen artiumExamen 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 1630...

. 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

CopenhagenCopenhagen is the capital and largest city of Denmark, with an urban population of 1,199,224 and a metropolitan population of 1,930,260 . With the completion of the transnational Øresund Bridge in 2000, Copenhagen has become the centre of the increasingly integrating Øresund Region...

and visit Ferdinand Degen and other mathematicians there. While in Copenhagen, Abel did some work on

Fermat's Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....

. Abel's uncle, Peder Mandrup Tuxen, lived at the naval base in

ChristianshavnChristianshavn is an artificial island neighbourhood located in Copenhagen, Denmark. It was founded in the early 17th century by Christian IV as part of his extension of the fortifications of Copenhagen. Originally it was laid out as an independent privileged merchant's town with inspiration from...

, Copenhagen, and at a ball there Niels Abel met Christine Kemp, his future fiancée. In 1824, Christine moved to

SonSon is an old town, and a former municipality. It is located in Vestby municipality in Akershus.- Overview :Son is located at the Oslofjord, 50 kilometres south of the Norwegian capital Oslo, just north of Moss and near the border of Østfold, and is located west of the village of Hølen. Before the...

, Norway to work as a governess and the couple got engaged over Christmas, 1824.

After returning from Copenhagen, Abel applied for a government

scholarshipA scholarship is an award of financial aid for a student to further education. Scholarships are awarded on various criteria usually reflecting the values and purposes of the donor or founder of the award.-Types:...

in order to visit top mathematicians in

GermanyGermany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

and

FranceThe French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

; but instead, he was granted 200 speciedaler yearly for two years, to stay in Cristiania 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). For, in 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals (now referred to as the

Abel–Ruffini theoremIn algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...

). 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

proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

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 .- History :...

.

In 1825, Abel wrote a personal letter to

King Carl Johan of Norway/SwedenCharles XIV & III John, also Carl John, Swedish and Norwegian: Karl Johan was King of Sweden and King of Norway from 1818 until his death...

requesting permission to travel abroad immediately. 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). The four 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

GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

in

GöttingenGöttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...

and then continue to Paris. However, when he got as far as Copenhagen he changed his plans. He wanted to follow his friends to

BerlinBerlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

instead, intending to visit Göttingen and Paris afterwards.

On the way, he visited the astronomer

Heinrich Christian SchumacherHeinrich Christian Schumacher was a German-Danish astronomer.-Biography:He was born at Bramstedt, in Holstein, and studied at Kiel, Jena, Copenhagen, and Göttingen. In 1810, he became adjunct professor of astronomy in Copenhagen...

in Altona, now a district of Hamburg. He then spent four months in Berlin, where he became well acquainted with

August Leopold CrelleAugust Leopold Crelle 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...

, who was then about to publish his mathematical journal,

Journal für die reine und angewandte MathematikCrelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik .- History :...

. 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

LeipzigLeipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...

and

FreibergFreiberg is a city in the Free State of Saxony, Germany, administrative center of the Mittelsachsen district.-History:The city was founded in 1186, and has been a center of the mining industry in the Ore Mountains for centuries...

to visit

Georg Amadeus Carl Friedrich NaumannGeorg Amadeus Carl Friedrich Naumann , also known as Karl Friedrich Naumann, was a German mineralogist and geologist. The crater Naumann on the Moon is named after him....

and his brother the mathematician August Naumann. In Freiberg Abel did brilliant research in the theory of functions, particularly:

ellipticIn complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

, 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. July 1826 Abel traveled on his own from Basel to Paris. Abel had sent most of his work to Berlin to be published in Crelles Journal, but he had saved what he regarded his most important work for the

French Academy of SciencesThe French Academy of 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...

, a theorem on addition of algebraic differentials. With the help of 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. 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 Crelles 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 Crelles Journal. But in mid-1828 he published, in rivalry with

Carl JacobiCarl Jacobi may refer to:* Carl Gustav Jacob Jacobi - Prussian mathematician and teacher* Carl Richard Jacobi - American author* Carl Jacobi - German architect...

, an important work on elliptic functions in

Astronomische NachrichtenAstronomische Nachrichten , one of the first international journals in the field of astronomy, was founded in 1821 by the German astronomer Heinrich Christian Schumacher. It claims to be the oldest astronomical journal in the world that is still being published...

in Altona.

### Death

While in

ParisParis is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

, Abel had contracted

tuberculosisTuberculosis, MTB, or TB is a common, and in many cases lethal, infectious disease caused by various strains of mycobacteria, usually Mycobacterium tuberculosis. Tuberculosis usually attacks the lungs but can also affect other parts of the body...

. For Christmas 1828, he traveled by sled to

FrolandFroland is a municipality in Aust-Agder county, Norway. It is part of the traditional region of Sørlandet. The administrative center is the village of Blakstad, which along with Osedalen form the main population center of the municipality....

to visit again his fiancée. He became seriously ill on the journey and, although a temporary improvement allowed the couple to enjoy the holiday together, died just two days before a letter arrived from August Crelle. All this time, Crelle had been searching for a new job for Abel in Berlin, and had actually managed to have him appointed a professor at a university. Crelle wrote to Abel on 8 April 1829 to tell him the good news, but it came too late.

## Mathematical work

Abel gave a proof of the

binomial theoremIn elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

valid for all numbers, extending

EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

's result which had held only for rationals. At age 19, he showed there is no general

algebraicAlgebraic may refer to any subject within the algebra branch of mathematics and related branches like algebraic geometry and algebraic topology.Algebraic may also refer to:...

solution for the roots of a

quintic equationIn mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as

group theoryThe history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry...

, which is invaluable not only in many areas of mathematics, but for much of

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

as well. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils." Abel said famously of

Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

's writing style, “He is like the fox, who effaces his tracks in the sand with his tail.”

## Legacy

The early death of this talented mathematician, of whom

Adrien-Marie LegendreAdrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

said "quelle tête celle du jeune Norvégien!" ("what a head the young Norwegian has"), cut short a career of extraordinary brilliance and promise. Under Abel's guidance, the prevailing obscurities of

analysisAnalysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle , though analysis as a formal concept is a relatively recent development.The word is...

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 .- History :...

, were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by

Ludwig SylowPeter Ludwig Mejdell Sylow was a Norwegianmathematician, who proved foundational results in group theory. He was born and died in Christiania ....

and

Sophus LieMarius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...

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 groupIn abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

,

abelian categoryIn 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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

, and

abelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and 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 functions...

).

On 6 April 1929, four Norwegian stamps were issued for the centenary of Abel's death. His portrait appears on the 500-

kronerThe krone is the currency of Norway and its dependent territories. The plural form is kroner . It is subdivided into 100 øre. The ISO 4217 code is NOK, although the common local abbreviation is kr. The name translates into English as "crown"...

banknoteA banknote is a kind of negotiable instrument, a promissory note made by a bank payable to the bearer on demand, used as money, and in many jurisdictions is legal tender. In addition to coins, banknotes make up the cash or bearer forms of all modern fiat money...

(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

AbelAbel is an ancient lunar crater that lies near the southeast limb of the Moon's near side. It is located to the south of the crater Barnard, at the northwest edge of the Mare Australe....

on the

MoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

was named after him. In 2002, the

Abel PrizeThe Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has often been described as the "mathematician's Nobel prize" and is among the most prestigious...

was established in his memory.

Mathematician

Felix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

wrote about Abel:

## See also

- 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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

- Abel equation
The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the formf = h\,\!or\alpha=\alpha+1\!and shows non-trivial properties at the iteration.-Equivalence:...

- Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

- Abel's identity
In mathematics, Abel's identity is an equation that expresses the Wronskian of two homogeneous solutions of a second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.The relation can be generalised to nth-order linear ordinary differential...

- Abel's inequality
In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case....

- Abelian means
- Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.-Theorem:...

- Abel transform
In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions...

- Abel transformation

## External links