Let be a non-constant polynomial over a field . Given a field extension over we say thatsplitsover iff where i.e. into linear factors. Furthermore, we say that is asplitting fieldof over iff i.e. is the smallest such field extension for which the polynomial splits over.

Returning back to your problem with in the field . Look at the first factor, , it has no zeros in by simply checking. Therefore this polynomial is irreducible. Form the factor ring which turns out being a field since is an irreducible polynomial, this is also a finite field with elements, so we will refer to it as . The mapping embeds in and therefore we can think of being contained in . Let and so , this means that is root of the polynomial . It is not hard to see that is another root of . Therefore, . Now the question is whether the second factor, , splits over . A little guessing shows that is a root of this polynomial because . And so the other root is easy to find which is which means and . Therefore, is a splitting field.