Splitting Field

Let be a Field and .
Then the splitting field of is the smallest field extension of
such that factorizes into linear factors.

Method

In general, to find the splitting field we do the following:
Factor into irreducible factors in .
Choose any nonlinear factor .
Construct the field extension .
In this new field, will be a root of and hence of
Continue the process until completely factors.

We are basically introducing a new root that satisfies at each step.

Lemma

Let be prime and be a Finite Field
Let be an irreducible polynomial of degree in
The splitting field of over is
(i.e. )

Proof (sketch)

Needs Galois Theory but basically, after adding one root to
we can check that the resulting field is and we can also check
that we have added every root.
This is because if , then etc.
(due to A Freshman’s Dream)
(I THINK AT LEAST)