Let be an algebraic number field with ring of integers . Let and be ideals in . Suppose and factor into prime ideals as and , where . Prove that . Prove also that .
Note that is an ideal, and so factors as a product of prime ideals as for some . Since , we have . In particular, for each . Similarly, since we have for each . Thus for each . Conversely, we have , so that . Thus , as desired.
Using the previous result, we have .