If $F/K$ is a field extension, $X^n-a \in K[x]$ is irreducible, $\alpha \in F$ is a root and $m \in \Bbb N$ a divisor of $n$, prove that degree of $\alpha^m$ on $K$ is $\frac n m$. What is the monic irreducible polynomial of $\alpha^m$ over $K$?
For the first part I have this:
K⊆K(α)⊆F (tower of fields) then [F:K]=[F:K(α)][K(α):K] degree of [F:K] is n, degree of [F:K(α)] is m then degree of [K(α):K] is n/m is this correct??
Thanks