Norwegian mathematician, born in Nedstrand, near Finnøy, where his father was rector. In he entered the Cathedral School in Christiania (now Oslo) and three years later he gave proof of his mathematical genius by his brilliant solutions of the original problems proposed by Bernt Holmboe. At that time, his father, Søren Georg Abel, a poor Protestant minister, died, and his family was in a difficult economic circumstances; however, a small pension from the Norwegian state allowed Abel to enter Christiania University in .
The first notable work of Abel was the proof of the impossibility of solving the quintic equation in radicals (Abel-Ruffini theorem). This paper was first published in in a difficult form, and after () in a more more elaborate form in the first volume of Crelle's Journal. Later, a state sponsorship allowed him to visit Germany and France in , and having visited the astronomer Schumacher in Altona, near Hamburg, he spent six months in Berlin, where he was well installed with August Leopold Crelle, owner of the mathematical journal. This project was strongly encouraged by Abel, who contributed much to the success of this venture. From Berlin, he went to Freiberg, where he developed his research on the theory of functions: elliptic, hyperelliptic, and a new class now known as abelian functions.
In Abel moved to Paris, and during months of his stay, he met the leading mathematicians of France; however, he was not appreciated, his work was little known and his modesty prevented him from publicize his research. In April , Crelle obtained a job for him in Berlin, but the letter that brought the offer came just two days after the death of Abel with tuberculosis in Froland Ironworks near Arendal.
The premature death of this talented mathematician, of which Adrien-Marie Legendre said "what head has this young Norwegian", interrupted an extraordinarily bright and promising career. His works, most of which originally appeared in Crelle's Journal, were edited by Holmboe and published in by the Swedish Government, and a more complete edition by Ludwig Sylow and Soplus Lie were published in . The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally writed with lower-case "a".
In , the Abel Prize was created in his honor.