## Oeuvres Compl�tes de N.H. Abel, Math�maticien, Avec Des Notes Et D�veloppements: 2

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- August 2nd 2018 by Wentworth Press

# Books by Niels Henrik Abel

## Niels Henrik Abel

**Date of birth:**August 05, 1802**Died:**April 06, 1829**Born:**in Finnoy, Norway.**Description:**Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for 250 years. He was also an innovator in the field of elliptic functions, discoverer of Abelian functions. Despite his incredible achievements, Abel was largely unrecognized during his lifetime; he made his discoveries while living in poverty and died at the age of 26.

Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "quelle tête celle du jeune Norvégien!" ("what a head the young Norwegian has!").Abel showed that there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Abel sent a paper on the unsolvability of the quintic equation to Carl Friedrich Gauss, who proceeded to discard without a glance what he believed to be the worthless work of a crank. Before Abel solved the problem, it had gone unsolved for over 250 years. He originally thought he had found a solution in 1821, which turned out to be wrong.As a 16 year old Abel gave a proof of the binomial theorem valid for all numbers, extending Leonhard Euler's result which had held only for rationals. Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions. While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). The paper was however, misplaced by Cauchy.While abroad Abel had sent most of his work to Berlin to be published in the Crelles Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. The theorem was put aside and forgotten until his death. While in Freiberg Abel did brilliant research in the theory of functions, particularly, elliptic, hyperelliptic, and a new class now known as abelian functions.In 1823 Abel wrote a paper titled "a general representation of the possibility to integrate all differential formulas" (Norwegian: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler). He applied for funds at the university to publish it. However the work was lost, while being reviewed, never to be found thereafter.Abel said famously of Carl Friedrich Gauss's writing style, "He is like the fox, who effaces his tracks in the sand with his tail."While in Paris, Abel contracted tuberculosis. During the Christmas of 1828, he traveled by sled to Froland to visit his fiancée. He became seriously ill on the journey and, although a temporary improvement allowed the couple to enjoy the holiday together, he died relatively soon after on 6 April 1829, just two days before a letter arrived from August Crelle. Crelle had been searching for a new job for Abel in Berlin, and had actually managed to have him appointed as a professor at the University of Berlin. Crelle wrote to Abel on 8 April 1829 to tell him the good news, but it came too late.

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