Niels Henrik Abel

Niels Henrik Abel (1802–1829) was a Norwegian mathematician whose work transformed the theory of elliptic functions and established that general quintic equations cannot be solved by radicals.

Abel–Ruffini theorem: The theorem that no formula using radicals solves every polynomial equation of degree five or higher. Abel supplied the decisive proof that the general quintic is not solvable by radicals.
Solvability by radicals: The expression of polynomial roots using arithmetic operations and extraction of roots. Abel’s theorem rules out this kind of expression for the general quintic.
University of Christiania: The institution founded in Oslo in 1811 as Norway’s first university, now the University of Oslo. Abel studied there after his mathematical talent was recognized.
Évariste Galois: A French mathematician whose theory of groups explains which polynomial equations are solvable by radicals. Galois gave a broader structural account of polynomial solvability after Abel’s impossibility proof.
Mémoire sur une propriété générale d’une classe très-étendue de fonctions transcendantes: Abel’s memoir presenting foundational results on elliptic functions and their transformations. This Paris memoir contains a central part of his work on elliptic functions.
Galois theory: The study of field extensions and symmetry groups used to characterize polynomial solvability. Abel’s impossibility result helped establish the problem that Galois theory later resolved structurally.
Quintic equation: A polynomial equation of degree five. The general quintic is the central case in Abel’s impossibility proof.
Christopher Hansteen: A Norwegian astronomer and physicist who taught at the University of Christiania. Hansteen supported Abel’s mathematical education and helped secure opportunities for him.
Carl Friedrich Gauss: A German mathematician whose work ranged across number theory, analysis, geometry, and astronomy. Abel’s elliptic-function research developed ideas connected to Gauss’s work on elliptic integrals.
Recherches sur les fonctions elliptiques: Abel’s major treatise on elliptic functions, published in 1827 and 1828. The treatise develops the theory that became one of Abel’s defining contributions.

Linked from 19 pages