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.
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.