does the polynomial split?

Question

Let $a \in \mathbb{C}$ such that $a^n $ is rational for some positive $n$.Let $ m$ be the smallest positive integer satisfying previous condition.Is it true that $ x^m- a^m$ splits in $\mathbb{Q}(a)[x]$?Thank you.

Answer 1

No, take $a=\sqrt[3]2$. Then $\mathbb Q(a)$ is a real field but $x^3-2$ does not split there because you need complex cube roots of unity.