Niels Henrik Abel

Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician 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 and the discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis.

Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!"

Life

Early life

Niels Henrik Abel was born prematurely 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. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the bailiff in Nedstrand in July and August of his year of birth.

Niels Henrik Abel's father, Søren Georg Abel, had a degree in theology and philosophy and served as pastor at Hesby Church on Finnøy. Søren's father, Niels's grandfather, Hans Mathias Abel, was also a pastor, at Gjerstad Church 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 and came to Norway in the 17th century.

thumb|right|alt=Postcard of Gjerstad church and rectory.|Postcard of Gjerstad church and rectory in 1890–95. The main building of the rectory was the same as when Abel lived here.

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 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. This led to a discovery in 1823 that a solution by formula to a fifth- or higher-degree equation was not necessarily possible.

Abel graduated in 1822. His performance was exceptionally high in mathematics and average in other matters.

Career

thumb|alt=From a notebook of Niels Henrik Abel|From the notebook of Niels Henrik Abel

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. 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. (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 was published in 1826 in the first volume of Crelle's Journal.

thumb|Memorial plaque unveiled in 2014 in Berlin where Abel lived 1825-26

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 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. Abel contributed seven articles to it in its first year.

From Berlin Abel also followed his friends to the Alps. He went to Leipzig and Freiberg to visit Georg Amadeus Carl Friedrich Naumann 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, 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. 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 in Altona.

Death

While in Paris, Abel had contracted tuberculosis. 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 telling Abel that Crelle had secured him an appointment as a professor at the University of Berlin.

Contributions to mathematics

Abel showed that there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations with the Abel-Ruffini theorem. To do this, he invented (independently of Galois) a branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics 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's result which had held only for rationals. Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions.

While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). The paper was, however, misplaced by Cauchy.

Legacy

Under Abel's guidance, the prevailing obscurities of analysis 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, were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by Ludwig Sylow and Sophus Lie was published in 1881.

The Abel Prize in mathematics was established in Abel's memory and named in his honour. Although it was originally proposed in 1899 to complement the Nobel Prizes, it was first awarded in 2003, while Selberg received an honorary Abel Prize the previous year.

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.

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 Municipality

File:Abelmonument.jpg|A monument for Abel at Frolands verk by Gustav Lærum

</gallery>

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

</gallery>

Notes

References

External links

Biographies and handwritten manuscripts from the Abel Prize website
Biography of Niels Henrik Abel
Translation of Niels Henrik Abel's "Research on Elliptic Functions" at Convergence
Famous Quotes by Niels Henrik Abel at Convergence
The Niels Henrik Abel mathematical contest, The Norwegian Mathematical Olympiad
Family genealogy