His first notable work was a proof of the impossibility of solving the quintic equation by radicals (see Abel-Ruffini theorem.) This investigation was first published in 1824 in abstruse and difficult form, and afterwards (1826) more elaborately in the first volume of Crelle's Journal. Further state aid enabled him to visit Germany and France in 1825, and having visited the astronomer Schumacher (1780-1850) at Altona, he spent six months in Berlin, where he became intimate with August Leopold Crelle, who was then about to publish his mathematical journal. This project was warmly encouraged by Abel, who contributed much to the success of the venture. From Berlin he passed to Freiberg, and here he made his brilliant researches in the theory of functions, elliptic, hyperelliptic, and a new class known as ''abelians being particularly studied.
In 1826 he moved to Paris, and during a ten months' stay he met the leading mathematicians of France; but he was little appreciated, for his work was scarcely known, and his modesty restrained him from proclaiming his researches. Pecuniary embarrassments, from which he had never been free, finally compelled him to abandon his tour, and on his return to Norway he taught for some time at Christiania. In 1829 Crelle obtained a post for him at Berlin, but the offer did not reach Norway until after his death near Arendal.
The early death of this talented mathematician, of whom Legendre said "quelle tête celle du jeune Norvegien!", cut short a career of extraordinary brilliance and promise. 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 Holmboe and published in 1839 by the Swedish government, and a more complete edition by Ludwig Sylow and Sophus Lie 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". (See abelian group and abelian category; also abelian variety.)
The Abel Prize is named after him.