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**eskimo343** Suppose $\displaystyle K/F$ is a field extension of degree $\displaystyle m$ and that $\displaystyle \alpha \in K$. Prove that for any integer $\displaystyle n$ such that $\displaystyle \text{gcd}(m, n)=1$, $\displaystyle F(\alpha)=F(\alpha^n).$

I was thinking initially in this problem to use the tower lemma but nothing seemed to work out after that. I do not know how to show that for any integer $\displaystyle n$ such that $\displaystyle \text{gcd}(m, n)=1$, $\displaystyle F(\alpha)=F(\alpha^n)$. Thanks in advance.