## Compute a minimal polynomial over QQ

Suppose $\theta$ is algebraic over $\mathbb{Q}$ with minimal polynomial $p(x) = x^3 - x^2 - x - 1$. ($p(x)$ is indeed irreducible since $p(x-1)$ is Eisenstein at 2.) Compute the minimal polynomial of $\eta = \theta^2+3$ over $\mathbb{Q}$.

Note that $\eta$ has degree at most 3 over $\mathbb{Q}$. Evidently, we have $\eta^2 = 8\theta^2 + 2\theta + 10$ and $\eta^3 = 52\theta^2 + 24\theta + 40$. Suppose $\eta^3 + a\eta^2 + b\eta + c = 0$; comparing coefficients, we have the following system of linear equations: $2a = -24$, $8a+b = -52$, and $10a+3b+c = -40$. Evidently then $\eta$ is a root of $t(x) = x^3 - 12x^2 + 44x - 52$. We can see using the rational root theorem that $t(x)$ has no linear factors over $\mathbb{Q}$, and thus is irreducible. So $t(x)$ is the minimal polynomial of $\eta$ over $\mathbb{Q}$.