Say that is irreducible then we need to show for any with (i.e. non-zero) we can find such that . Now since are relatively prime since is irreducible it means there exists so that and so i.e. .

Now you try proving the converse by assuming that is reducible.

How can I build a field with nine elements?

Define by and . Now show this extends to a function which is in fact a field isomorphism.Prove that