
Chapter 8
The Idea Behind Galois Theory
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 prob-
lems 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 ∈C[t],
he associated a group ...