## Every algebraic integer divides a rational integer

Prove that every algebraic integer divides a rational integer. (Here we mean divisibility in the extended sense that $\beta|\alpha$ if $\alpha/\beta$ is an algebraic integer, not necessarily in a given field.)

Let $\alpha$ be an algebraic integer with conjugates $\alpha_1, \ldots, \alpha_k$. (Recall that the $\alpha_i$ are also algebraic integers.) Let $N(\alpha)$ denote the norm of $\alpha$.

By Theorem 7.1 in TAN, $N(\alpha)$ is a rational integer. By definition, $\alpha|N(\alpha)$.