Abel, Niels Henrik (1802–1829)

Norwegian mathematician. He demonstrated that the general quintic equation ax5 + bx4 + cx3 + dx2 + ex + f = 0 could not be solved algebraically. Subsequent work covered elliptic functions, integral equations, infinite series, and the binomial theorem.

Abel was born on Finnöy, a small island near Stavanger, and studied in Oslo. In 1823 he provided the first solution in the history of mathematics of an integral equation and published a paper demonstrating that a radical expression to represent a solution to fifth- or higher-degree equations was impossible.

In 1825, Abel went to Berlin, where he met Leopold Crelle (1780–1855), a privy councillor and engineer much taken with problems in mathematics. Together they brought out the first issue of Crelle's Journal, which was to become the leading 19th-century German organ of mathematics. A year later Abel moved on to Paris, where he wrote ‘Mémoire sur une propriété générale d'une classe très-étendue de fonctions transcendantes’. It dealt with the sum of the integrals of a given algebraic function and presented the theorem that any such sum can be expressed as a fixed number of these integrals with integration arguments that are algebraic functions of the original arguments.

Abel transformed the theory of elliptic integrals by introducing elliptic functions, and this generalization of trigonometric functions led eventually to the theory of complex multiplication, with its important implications for algebraic number theory. He also provided the first stringent proof of the binomial theorem. A number of useful concepts in modern mathematics, notably the Abelian group and the Abelian function, bear his name.