A Dedekind domain with finitely many prime ideals is a UFD

Let be a Dedekind domain with finitely many prime ideals. Then every prime ideal is principal, whence is a PID and in particular a UFD. #m/thm/ring

Proof

Let enumerate all prime ideals in . By a similar construction to that in the proof of Ideals of a Dedekind domain need at most two generators, we can choose for each a which is not in or for via the Chinese remainder theorem for rings. It follows that .

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