Divisibility among rational integers in an algebraic number field

Let E = \mathbb{Q}(\theta) be an algebraic extension of \mathbb{Q} and let a,b \in \mathbb{Z}. Recall that if \alpha,\beta \in E are algebraic integers, we say that \beta|\alpha if the quotient \alpha/\beta is an algebraic integer. Prove that b|a in E if and only if b|a in \mathbb{Q}.


If b|a in \mathbb{Q}, then a/b is a rational integer. Certainly a/b \in E is an algebraic integer; so b|a in E.

Now suppose b|a in E. Since a/b is an algebraic integer and a rational number, a/b is a rational integer. So b|a in \mathbb{Q}.

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