Let $F$ be a field and $E=F(a)$, where the min poly of a has degree n over F. What are the conditions on $m\in \mathbb{Z}$ so that $E=F(a)=F(a^m)$

Question

So far I've been playing around with different techniques. I tried to show double inclusion to get the equality in hopes of coming across a needed condition on m, but was unsuccessull. I also worked with degrees and only got this far:

$[F(a^m):F]=[F(a^{m-1}):F(a)][F(a):F]$ and since we know that $[F(a):F]=n$ and we want $[F(a^m):F]=n$ as well, then this would imply that $[F(a^{m-1}):F]=1$

But now I'm stuck and unsure where to take it from here, or if I've even taken the correct approach to begin with.

Any tips would be greatly appreciated.