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.

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