Niels Henrik Abel - Mathematical Work

Mathematical Work

Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had held only 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." 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.”