Calculating the minimal polynomial of linear combinations of numbers

Question

I am trying to calculate the minimal polynomial of $\theta^2/2$, where $\theta$ is a root of $X^3-2X-2$ (I'm trying to work out if $\theta^2/2$ is an algebraic integer in $\mathbb{Q}(\theta)$). I have approached this the same way I approached $\theta/2$, writing $\theta^2/2$ as a linear combination of $1,\theta^2/2, (\theta/2)^2$, and $(\theta^2/2)^3$, but i have gotten stuck. Some help would be appreciated!

Answer 1

Divide $a (\theta^2/2)^3 + b (\theta^2/2)^2 + c (\theta^2/2) + d$ by $\theta^3 - 2 \theta - 2$. The remainder will be a polynomial of degree $2$ in $\theta$, which must be $0$. Solve the system of three equations for the coefficients.