Are these polynomials irreducible over $\Bbb Z$?

Question

I have received these problems and I'm not sure where to start:

Are these polynomials irreducible in $\Bbb Z[X]$ ?

a) $X^{2^n} + 1$ where $n$ is a positive integer.

b) $X^{p-1} + X^{p-2} + \cdots + X + 1 $ where $p$ is prime.

I know how to apply Eisenstein's Criterion, Gauss' Lemma and the Reduction Criterion but none of these seem to apply.

If anyone could explain any of these to me or give me a clue as to how to get started I'd be grateful.

Answer 1

These can all be solved with the same method: to show $f(x)$ irreducible, apply Eisenstein to $f(x+1)$.