ABSTRACT
We now return to our goal of understanding whether the roots of an irreducible polynomial over a field can be obtained by elementary algebraic computations. In Chapter 45 we constructed the unique splitting field for such a polynomial, inside of which such computations must occur. In the present chapter we will look closely at what sort of field extension the splitting field must be. We will use group theory to do this. In Chapters 22 and 23 we saw how geometry could be illuminated
by considering functions leaving geometric properties fixed; we thus obtained groups of symmetries. Here we will illuminate field extensions (and splitting fields in particular) by considering functions leaving field properties fixed; we will thus obtain groups of automorphisms called Galois groups.
