TITLE: Irreducibility of $X^n-a$
QUESTION [6 upvotes]: Let ${\mathbb K}$ be a subfield of ${\mathbb C}$. Let 
$a\in{\mathbb K}$ such that $X^d-a$ has no root in ${\mathbb K}$,
for any divisor $d>1$ of $n$. Does it follow that $X^n-a$ is irreducible
over $\mathbb K$ ?

REPLY [6 votes]: Here is the general theorem on such
Theorem $\ $ Suppose $\rm\:c\in F\:$ a field,  and $\rm\:0 < n\in\mathbb Z\:.$
$\rm\quad x^n - c\ $ is irreducible over $\rm\:F \iff c \not\in F^p\:$ for all primes $\rm\:p\: |\: n\:$ and $\rm\ c\not\in -4\:F^4\:$ when $\rm\: 4\ |\ n\:. $
A proof is in many Field Theory textbooks, e.g. Karpilovsky, Topics in Field Theory, Theorem 8.1.6.