## Show that a given algebraic integer is a unit in the ring of integers in the field extension it generates

Let $\alpha$ be a root of $p(x) = x^{17} - 43x^6 + 1$. Show that, as ideals in the ring $\mathcal{O}$ of integers in $\mathbb{Q}(\alpha)$, $(\alpha) = \mathcal{O}$.

Note that $p(x) = h\prod q_i(x)$ is a product of the minimal polynomials of its roots. Since $p(x)$ is monic, in fact $h = 1$. Since the constant coefficient of $p(x)$ is 1, the constant coefficient of the minimal polynomial $q(x)$ for $\alpha$ over $\mathbb{Q}$ is 1. That is, $N(\alpha) = 1$. By Theorem 7.3 in TAN, $\alpha$ is a unit; hence $(\alpha) = \mathcal{O}$.