a result about minimal polynomial and field extension

Question

For $K$ a field and $|K(\alpha):K| = 5$, why do we have $K(\alpha^2) = K(\alpha)$?

Answer 1

You have the extensions $ K \subseteq K(\alpha^2) \subseteq K(\alpha)$ and therefore regarding the degrees:

$$[K(\alpha):K(\alpha^2)][K(\alpha^2):K]=[K(\alpha):K]=5.$$

Therefore $[K(\alpha^2):K]\in \{1,5\}$. $[K(\alpha^2):K]=1$ can’t be as it means that $\alpha^2 \in K$ in contradiction with the degree of $\alpha$ over $K$. Hence $[K(\alpha^2):K]=5$ and the conclusion.