I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (thus, if I understood everything correctly, $a$ is a root of unity and $K$ is actually a cyclotomic field, hence the title of the question).
It's not difficult to show that $x^4+1 = (x-a)(x^3 + ax^2 + a^2x + a^3)$ in $K[x]$, using the fact that $a^4+1=0$. But I have no clue about how to show that the factors are irreducible. I thought about applying Eisenstein's criterion, but then got stuck over whether $a$ is a prime in the ring of algebraic integers of $Q(a)$. Is this the correct way?
It'd be nice to know a general procedure for this type of question.