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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?

edited Nov 1 at 21:39

Gerry Myerson
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asked Sep 5 at 5:25

LeBlanc
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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 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 at 13:25

Will Sawin · Accepted Answer · 2012-11-01 20:23:03Z UTC

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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.

edited Nov 1 at 20:23

answered Sep 5 at 5:28

Will Sawin
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Ah, silly me. Thank you :) – LeBlanc Sep 5 at 5:38

Is every countable Dedekind domain the ring of integers of some number field?

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