suppose b is a root of .
let k be any element of .
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 .