# Abel, Niels Henrik

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## Abel, Niels Henrik

(nēls hĕn`rĭk ä`bəl), 1802–29, Norwegian mathematician. While a student at the Univ. of Christiania (Oslo) he did fundamental work on the integration of functional expressions and proved the impossiblity of representing a solution of a general equation of fifth degree or higher by a radical expression. He investigated generalizations of the binomial theorem, pioneered in the general theory of elliptic functions, and showed that elliptic functions are a generalization of trigonometric functions. Commutative groups**group,**

in mathematics, system consisting of a set of elements and a binary operation

*a*+

*b*defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e.

**.....**Click the link for more information. are also called Abelian groups in his honor. He died of tuberculosis at the age of 26, leaving contributions that rank him as one of the greatest mathematicians of the 19th cent.

### Bibliography

See O. Ore, *Niels Henrik Abel: Mathematician Extraordinary* (1957, repr. 1973).

## Abel, Niels Henrik

Born Aug. 5, 1802; died Apr. 6, 1829. Norwegian mathematician and one of the outstanding mathematicians of the 19th century.

Abel was born near Stavanger into the family of a pastor and studied in Kristiania (Oslo). His extraordinary mathematical abilities began to appear when he was 16. From 1825 to 1827 he traveled through Europe, establishing friendships with many well-known mathematicians. Abel achieved no recognition in his native land during his lifetime; he lived in poverty and died, near Arendal, from tuberculosis. A monument was erected to him in Oslo in 1908.

Abel’s work exerted a great influence on the development of mathematics as a whole, for it resulted in a number of new mathematical disciplines—the Galois theory and the theory of algebraic functions. It also led to the general recognition of the theory of functions of a complex variable. Abel’s first investigations were in algebra. He showed in 1824 and 1826 that while algebraic equations of degree higher than the fourth are insoluble in radicals in the general case, there are particular types of equations that are solvable in radicals; the groups that are associated with these equations are called abelian groups. In integral calculus, he studied integrals of algebraic functions, which became known as abelian integrals. One of the founders of the theory of elliptical functions, Abel was also very important in the substantiation of mathematical analysis. He systematically underscored the need for using only convergent series and in 1826 investigated the region of convergence of a binomial series for complex values of the variables and the properties of functions that can be represented by power series. In addition, he wrote the first study of integral equations. Abel’s works also had a certain impact on the theory of interpolation of functions, the theory of functional equations, and number theory.

### WORKS

*Oeuvres completes*, vols. 1–2. Kristiania, 1881.

### REFERENCE

Ore, O.*Zamechatel’nyi matematik Nil’s Genrik Abel’*. Moscow, 1961. (Translated from English.)

S. B. STECHKIN