By ^P1 can be reduced down to a product of prime cyclotomic polynomials which we know have integer coëfficients.

Now suppose towards contradiction that is reducibile. Then its roots with are divided among the factors, so choose a root of one irreducible monic factor , so that another root for a prime is not a root of . Write

where are monic. Then is the minimal polynomial of over and .

It follows that is a root of , and hence , whence

for . Letting underling denote the projection , and invoking the Frobenius automorphism we have

so and have a nontrivial common factor in . It follows

so has a multiple factor.

This implies that is inseparable. On the other hand, its derivative is nonzero (since ), contradicting ^P1. Therefore must be irreducible.

If , then every th root of unity is also an th root of 1, which holds in particular for every primitive th root of unity.

On the other hand, every generates a subgroup , where if then . Thus every is a primitive th root of unity for some .

Therefore the set of th roots of unity eauals the union of the sets of primitive th roots of unity as ranges over factors of , whence follows ^P1.