Prove that for any prime and any nonzero , is irreducible and separable over .
Note that , so that and are relatively prime. So is separable.
Now let be a root of . Using the Frobenius endomorphism, , so that is also a root. By induction, is a root for all , and since has degree , these are all of the roots.
Now , and in particular the minimal polynomials of and have the same degree over – say . Since is the product of the minimal polynomials of its roots, we have for some . Since is prime, we have either (so that , a contradiction) or , so that itself is the minimal polynommial of , hence is irreducible.