Let $E/F$ a field extension and $f=X^n-a \in F[X]$ irreducible polynomial and and $m \in \mathbb{N} $ such that $m|n$ and $\rho$ a root of $f$.
Prove that the minimal polynomial of $\rho^m$ is $g=X^{n/m}-a$
One thought is to prove that $g$ is irreducible over $F$.If $g$ was not irreducible then we know that $irr(\rho^m,F)|g$
Can someone help me to reach a contradiction or give me a hint to solve it straight forward without contradiction?
Thank you in advance!
Hint: Note that $f(X)=g(X^m)$. Suppose you could factor $g(X)=p(X)q(X)$. What would this tell you about $f(X)$?