Suppose that we have a nonzero polynomial f(x) with coefficients in Q (or more generally any subfield of C). Then we know by the fundamental theorem of algebra that we can factorise our polynomial f(x) as a product of linear factors (i.e. find the roots of f(x)) in the complex numbers).

But what do we do if we have a field which is not a subfield of C, such as a finite field? We still want to be able to find a bigger field in which a polynomial will factorise. That is what splitting fields are for.

Definition Let M be a field extension of K.

  1. A nonzero polynomial f(x) in K[x] splits over M if there exists a,bi in M such that f(x)=a(x-bn)...(x-b0).
  2. M is called a splitting field for f(x) if f(x) splits over M and if whenever M is a field extension of L such that f(x) also splits over L then M=L.

In other words, a splitting field is a field where you can find all the roots of your polynomial and it is as small as possible. It is a nontrivial fact that if a splitting field exists then it is unique up to isomorphism. Here is a proof of the uniqueness of splitting fields.

I will sketch the proof that splitting fields always exist. Suppose that f(x) is irreducible. Then we can consider the quotient ring K[x]/f(x)K[x]. It is quite easy to prove that this ring is also a field and is a field extension of K. Furthermore, if we write a for the image of x then in this ring we have that f(a)=0. Thus, we are able to make a field extension of K which contains at least one zero of f(x). By repeating this process we can create a splitting field.