Suppose that we have a nonzero polynomial f(x)
with coefficients in
(or more generally any subfield
). Then we know by the fundamental theorem
that we can factorise our polynomial f(x)
product of linear factors (i.e. find the roots of f(x)
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.
Let M be a field extension of K.
A nonzero polynomial
f(x) in K[x] splits over M if there exists
a,bi in M such that
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
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.