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

It is not an exercise problem although it looks very simple. Does anyone knows the answer?

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

It is not an exercise problem although it looks very simple. Does anyone know the answer?