I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a lower bound on the degrees of the factors, by saying: there is a factor of size at least... ?
I know the result which says that $x^n-a$ is irreducible if $a$ is not in $\mathbb{Q}^p$ for some $p$ that divides $n$ and if we assume, in the case where $4|n$, that $a\not\in -4\mathbb{Q}^4$. [Lang, Algebra, VI, §9, Theorem 9.1] What happens in the other cases?
PS: This is related to another question I asked, which has still no answer: Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$
