I saw this exercise in an old exercise sheet and it's driving me nuts. There it is:
For the first part, I think I have to apply the tower theorem and then I also know a property that says that the degree of the minimal polynomial of $a \in K$ over $F$ divides $[K:F]$ and may be useful.
It may be $x^p -a$ being $p=n/m$ but I don't know how to formalize it.
Can someone give me a helping hand?
Yes you can use the tower theorem:
Since $x^n-a$ is irreducible, it is the minimal polynomial of $b$ over $F$. So $[F(b):F]=n$. Now you can check that $x^m-b^m \in F(b^m)[x]$ is the minimal polynomial of $b$ over $F(b^m)$, hence $[F(b):F(b^m)]=m$. Using tower law, we get $[F(b^m):F]=p=\dfrac{n}{m}$ and since $x^p-a \in F[x]$ is of degree $p$ and has a solution $b^m$, that is the minimal polynomial of $b^m$ over $F$.
