Born: 5-Aug-1802

Died: 6-Apr-1829

At an early age, Niels Abel studied the works of the greatest mathematicians, found flaws in their proofs, and resolved to reprove some of these theorems rigorously. He was the first to fully prove the general case of Newton's Binomial Theorem, one of the most widely applied theorems in mathematics. Perhaps his most famous achievement was the (deceptively simple) Abel's Theorem of Convergence (published posthumously), one of the most important theorems in analysis; but there are several other Theorems which bear his name. Abel also made contributions in algebraic geometry and the theory of equations.

Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had only held for rationals. At age 19, he showed 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. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils."[4] 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.

Inversion (replacing y = f(x) with x = f-1(y)) is a key idea in mathematics (consider Newton's Fundamental Theorem of Calculus); Abel developed this insight. Legendre had spent much of his life studying elliptic integrals, but Abel inverted these to get elliptic functions, which quickly became a productive field of mathematics, and led to more general complex-variable functions, which were important to the development of both abstract and applied mathematics.

While in Paris, Abel had contracted tuberculosis. For Christmas 1828, he traveled by sled to Froland to visit again his fiancée. He became seriously ill on the journey and, although a temporary improvement allowed the couple to enjoy the holiday together, died just two days before a letter arrived from August Crelle. All this time, Crelle had been searching for a new job for Abel in Berlin, and had actually managed to have him appointed a professor at a university. Crelle wrote to Abel on 8 April 1829 to tell him the good news, but it came too late.

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