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Is every countable Dedekind domain the ring of integers of some number field? I tried googling different keywords, but did not find anything. Does anyone know of research in this area?

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Dear LeBlanc, In addition to the counterexamples of Will Sawin, note that any localization of a ring of integers is again a countable Dedekind domain. Regards, –  Emerton Sep 5 '12 at 6:37
    
As a general note, the ring of integers of a number field are a quite special type of Dedekind domain. E.g., for them each ideal class contains a prime ideal (and in suitable senses even the same number of them). In general this is not true. –  quid Sep 5 '12 at 13:25

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Nope. $\mathbb F_p[x]$, $\mathbb Q[x]$, and all other affine algebraic curves over countable fields, are countable Dedekind domains. None are the ring of integers of a number field.

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Ah, silly me. Thank you :) –  LeBlanc Sep 5 '12 at 5:38

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