Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits over $E$.
It's easy to see that $f(x)$ divides $f(x^2)$ in $F[x]$, and that all the roots of $f(x)$ are roots of unity by an easy induction (if $f(x) \neq x$). But I don't know how to proceed to get the conclusion.