If the degree of $\alpha$ is prime, then it equals the degree of $\alpha^q$ for every other prime $q$?

Question

Let $p$ and $q$ be two distinct prime integers. For a field $F$, assume that $\deg(\alpha, F)=p$. Is it necessarily true that $\deg(\alpha^q, F)=p$? Is there any counterexample?

Answer 1

It's not necessarily true, try $p=2$, $q=3$, $F=\mathbb{Q}$ and $\alpha=e^{2\pi i/3}$.

Answer 2

Somewhat generalizing the answer of Jarek Kuben, let $p,q$ be primes with $q-1=pr$ for some integer $r$, let $\alpha=e^{2\pi i/q}$ be a nonreal $q$th root of unity, let $K={\bf Q}(\alpha)$. Then $K$ is of degree $pr$ over the rationals, and is cyclic, so it has a subfield $E$ of degree $r$ over the rationals. Then $\alpha$ is of degree $p$ over $E$, but $\alpha^q=1$ is of degree 1 over $E$.