Editing Niels Henrik Abel (section)

=== Career ===
[[File:Abel-kladd.jpg|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" (''[[Dano-Norwegian#As a written language|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.<ref name="SNL-bio" />

[[File:Christine Kemp.jpg|thumb|Christine Kemp]]

In mid-1823, Professor Rasmussen gave Abel a gift of 100 [[speciedaler]] so he could travel to [[Copenhagen]] 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]], Copenhagen, and at a ball there Niels Abel met Christine Kemp, his future fiancée. In 1824, Christine moved to [[Son, Norway|Son]], Norway, to work as a governess and the couple got engaged over Christmas.<ref name="SNL-bio" />

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.<ref name="SNL-bio" /> 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''<ref>{{Cite web |url=http://www.abelprize.no/c54178/binfil/download.php?tid=53608 |title=Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré |access-date=28 July 2015 |archive-date=8 August 2021 |archive-url=https://web.archive.org/web/20210808172819/https://www.abelprize.no/c54178/binfil/download.php?tid=53608 |url-status=dead }}</ref> (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 [[Mathematical proof|proof]] was published in 1826 in the first volume of ''[[Crelle's Journal]]''.
[[File:Gedenktafel Am Kupfergraben 4a (Mitte) Niels Henrik Abel.jpg|thumb|Memorial plaque unveiled in 2014 in Berlin where Abel lived 1825-26]]
In 1825, Abel wrote a personal letter to [[Charles XIV John of Sweden|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 [[Carl Friedrich Gauss|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.<ref name="SNL-bio" /> 

On the way, he visited the astronomer [[Heinrich Christian Schumacher]] in Altona, now a district of Hamburg. He then spent four months in Berlin, where he became well acquainted with [[August Leopold Crelle]], who was then about to publish his mathematical journal, ''[[Crelle's 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]] and [[Freiberg, Saxony|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 function|elliptic]], [[Hyperelliptic function|hyperelliptic]], and a new class now known as [[abelian function]]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]], 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.{{By whom|date=June 2020}} 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 Gustav Jacob Jacobi|Carl Jacobi]], an important work on elliptic functions in ''[[Astronomische Nachrichten]]'' in Altona.