It suffices to prove the case where is not principal. Since A Dedekind domain admits UFI, we have

for distinct prime ideals and exponents . Choose some and consider . Since A Dedekind domain is a CDR, we have

where are distinct from each other and the and with .

We seek

For each , let , and for each , let . By the Chinese remainder theorem for rings, we have a surjective homomorphism

Then any

will do the trick.

We claim . Note

and also where is not divisible by any or . Hence .