For what values of $n > 1$ and a is $x^n - a$ irreducible in $\mathbb Q[x]$?

Question

$$x^n-a$$

So $n$ is any integer greater than 1, and $a$ is any integer. $a$ being any integer is where I am running into trouble. I have already shown and worked out a proof for this being irreducible when a is prime. Now I am just working through examples, trying to figure out a pattern.

The exercise is asking for application of Eisenstein's Criterion only, so I am assuming that the value of n should have no effect on the irreducibility.

Answer 1

Theorem $\ $ Suppose $\,F\,$ is a field and $\:a\in F\:$ and $\:0 < n\in\mathbb Z.\ $ Then

$\ \ \ x^n - a\ $ is irreducible over $F \iff a \not\in F^{\large p}\,$ for all primes $\,p\mid n,\,$ and $\ a\not\in -4F^4$ when $\: 4\mid n $

For a proof see e.g. Karpilovsky, Topics in Field Theory, Theorem 8.1.6, excerpted below, or see Lang's Algebra (Galois Theory).