Galois Extensions, An Introduction

Introduction

For his ground-breaking work in groups, Galois is widely considered to be the father of group theory. But the story doesn't end their. Galois was able to connect field extensions to groups, so that subfields and subgroups correspond. This forms a bridge between group theory and field theory, and theorems in one arena can sometimes be brought to bear in the other.

The connection between groups and fields has since been generalized. If two separate structures can be broken down into a hierarchy of substructures, subfields and subgroups for example, and a map carries substructures to substructures and preserves the partial ordering, that map is an instance of galois theory. Thus Galois created a powerful branch of mathematics that is used to solve everything from quintic polynomials to Fermat's last theorem.

We'll never know what else he could have accomplished, had he lived past the age of 20. Unfortunately he died in a duel, a lover's quarrel. He was gifted with unimaginable mathematical genius, but his judgment, and marksmanship, were lacking. Read more about this amazing, troubled man.