Splitting polynomials and fields: Definitions and motivation

Let F be any field, and f be a monic polynomial of degree n in F[X]. This polynomial is said to split in F if it factors completely, i.e., factors as a product of n linear factors x-ri. The ri are then the roots of f, that is, the solutions of the equation f(x)=0. If K is some extension of F, we likewise say f splits in K if can be written as a product (x-r1)(x-r2)...(x-rn) of n linear factors in K[X]. Clearly f then splits also in F(r1,r2,...,rn), the subfield of K generated by the roots. We say that K is a splitting field of f over F if f splits in K and K=F(r1,r2,...,rn).

We shall show that every polynomial has a splitting field K, which is unique up to automorphisms. When the polynomial is separable, the automorphisms of K that act trivially on F (a.k.a.  the ``automorphisms of K/F'') will constitute the Galois group Gal(K/F) of K over F.

Galois himself didn't have to worry about the construction of splitting fields, because he worked in a context in which F is either contained in an algebraically closed field (namely C), or F is finite, in which case we shall see that F(r1) is already a splitting field because the other roots are rq, rq2, rq3, ..., rqn-1, where q=|F|. In general we can construct a splitting field by adjoining roots one at a time, but then we have to prove that the resulting field does not depend on the order in which we adjoined the roots.