## A fact about norms and traces of algebraic integers

Let $K$ be an algebraic number field with ring of integers $\mathcal{O}$. Let $\alpha,\beta \in \mathcal{O}$ be nonzero. Prove that $N(\alpha)\mathsf{tr}(\beta/\alpha) \in \mathbb{Z}$.

Recall from Lemma 7.1 that $N(\alpha)$ is a rational integer, so that $N(\alpha)\mathsf{tr}(\beta/\alpha) = \mathsf{tr}(N(\alpha)\beta/\alpha)$. Now $N(\alpha)/\alpha$ is an algebraic integer, so that $N(\alpha)\beta/\alpha$ is an algebraic integer. Thus $\mathsf{tr}(N(\alpha)\beta/\alpha)$ is a rational integer.