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