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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.

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