TITLE: Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$
QUESTION [9 upvotes]: Given that $\alpha$ is a root (in the field extension) of the irreducible polynomial $X^4+X^3-X+2\in\mathbb{Q}[X]$, I have to find the minimal polynomial of $\alpha^2$. I am thinking about this for a while, but I can't find it. I need some hints. Thank you.

REPLY [5 votes]: Look up the Dandelin-Graeffe iteration. In short, $p(x)\cdot p(-x)$ is a polynomial in $x^2$, say $q(x^2)=p(x)\cdot p(-x)$. So if $p(\alpha)=0$, then $q(\alpha^2)=0$.
This gives the same result as the answer by Zarrax, however $q$ need not be irreducible.