suppose b is a root of .

let k be any element of .

then .

we then have p distinct roots of , so this IS the splitting field of .

now consider given by .

the assignment is a group homomorphism .

since , we know that or:

. the former implies that ,

that is, that b is in . since a ≠ 0, this cannot be true, so ,

which in turn means is irreducible over .