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).

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