## Compute the minimal polynomial of a given element over the rationals

Compute the minimal polynomial of $1 + \sqrt[3]{2}$ over $\mathbb{Q}$.

Let $\eta = 1 + \sqrt[3]{2}$. Evidently, $\eta^2 = 1 + 2 \sqrt[3]{2} + \sqrt[3]{2^2}$ and $\eta^3 = 3 + 3 \sqrt[3]{2} + 3 \sqrt[3]{2^2}$. Evidently, $\eta^3 - 3\eta^2 = -3 \sqrt[3]{2}$, so that $\eta^3 - 3\eta^2 - 3 = -3\eta$. Hence $\eta$ is a root of $p(x) = x^3 - 3x^2 + 3x - 3$. This polynomial is Eisenstein at 3, hence irreducible over $\mathbb{Q}$. By this previous exercise, $p(x)$ is the minimal polynomial of $\eta$ over $\mathbb{Q}$.