This is a part of a question from a Berkeley prelim exam and I would appreciate a hint, since I can't see a promising approach to this.
Let $p$ be an irreducible polynomial over the rationals with a nonzero complex root $a$. Suppose that $a^2$ is also a root of $p$. How would one conclude from this that for any root $b$ of $p$, $b^2$ is also a root of $p$?