Abstract Algebra – Minimal Polynomial of ?^2 Given the Minimal Polynomial of ?

Question

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.

Answer 1

You have $\alpha^4 + 2 = \alpha - \alpha^3$, and squaring both sides gives a polynomial of degree $4$ satisfied by $\alpha^2$. (All the powers of $\alpha$ appearing will be even.) Then show it is the minimal polynomial for $\alpha^2$.