Welcome to MathOverflow

Question

1

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
15.7k●1●47●87

asked Sep 5 at 5:25

LeBlanc
63●5

3

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

11

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
18.5k●1●24●48

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

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

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.