Let $K(\alpha)/K$ be a field extension of degree 4 such that $\alpha^2$ is not a root of the minimal polynomial of $\alpha$ over $K$. Find the degree of $K(\alpha^2)/K$.
So far I've been able to show two very basic things: such degree divides 4 and it is not 1. Hence, it is 2 or 4. Through examples, it seems to be the case that it is 2, but so far I've been unable to prove it. Any thoughts?
Indeed, if the degree was 1, we would have that $\alpha^2\in K$ and the minimal polynomial of $\alpha$ over $K$ would divide $x^2-\alpha^2$ which, in turn, would imply that the degree of the extension $K(\alpha)/K$ would be 1 or 2, in contradiction with the hypothesis.
I suspect I have to assume that the degree is 4 and conclude that $\alpha^2$ is a root of the minimal polynomial of $\alpha$ over $K$, somehow using that, in this case, $K(\alpha)=K(\alpha^2)$ and, in particular, $\alpha\in K(\alpha^2)$.