This is the statement I am trying to prove or disprove:

Let where for some prime number p and some positive integer n. Then has exactly n irreducible factors over the integers.

I can see how such polynomials could be factored into n factors but I am having difficulty proving or disproving that the factors are irreducible.

For instance, if p is 2 and n is 4 we have:

Clearly these factors are irreducible.

If p is 3 and n is 3 we have:

If p is 5 and n is 3 we have:

So I see the pattern that produces these factors, but I don't know how to show that all of them are irreducible if they are.