Let A be a DVR in its field of fractions F, and F′⊂F a subfield. Then is it true that A∩F′ is a DVR in F′? I can see that it is a valuation ring of F′, but how do I show, for example, that A∩F′ is Noetherian?

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soanooooo40 · Accepted Answer

Step 1This is not necessarily true. For instance, you could have A=k[[t]] for a field k and F′⊂k((t)) equal to k. Then A∩F′=F′ and a field is often not considered a DVR.But this is the only thing that can go wrong. In general, if a domain has a nontrivial valuation taking values in the nonnegative integers, such that the elements of valuation zero are units, then it is a DVR (and noetherianness follows - for instance, one checks that the only ideals occur as x:x has valuation at least n.)

Let A be a DVR in its field of fractions F, and F'\subset F a subfield. Then is it true that P

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