Galois Theory, 5th Edition

Chapter 8The Idea behind Galois Theory

DOI: 10.1201/9781003213949-8

Having satisfied ourselves that field extensions are good for something, we can focus on the main theme of this book: the elusive quintic, and Galois's deep insights into the solubility of equations by radicals. We start by outlining the main theorem that we wish to prove, and the steps required to prove it. We also explain where it came from.

We have already associated a vector space to each field extension. For some problems this is too coarse an instrument; it measures the size of the extension, but not its shape, so to speak. Galois went deeper into the structure. To any polynomial $p\in ℂ[t]$, he associated a group of permutations, now called the Galois group of p in his honour. ...