Timeline for When is a local subring of a number field a valuation ring?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2014 at 3:50 | comment | added | Neil Epstein | @abx Sorry for my typo (brain-o?), and thanks for catching it. | |
| Aug 20, 2014 at 19:41 | vote | accept | abcdxyz | ||
| Aug 20, 2014 at 16:44 | review | Close votes | |||
| Aug 21, 2014 at 0:45 | |||||
| Aug 20, 2014 at 16:04 | answer | added | paul garrett | timeline score: 1 | |
| Aug 20, 2014 at 13:18 | history | edited | abcdxyz | CC BY-SA 3.0 |
added 193 characters in body
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| Aug 20, 2014 at 6:30 | comment | added | abcdxyz | The problem with the above criterion is that it is probably quite a good criterion to show that something is not a valuation ring, but not very good at showing something is a valuation ring (which is what I need). | |
| Aug 20, 2014 at 5:29 | comment | added | abx | Bad example : \mathfrak{m} is in fact generated by $1+\sqrt{-5}$ (note that $3$ is invertible in $R$, and $2.3=(1+\sqrt{-5})(1-\sqrt{-5}))$. Replace $\sqrt{-5}$ by $\sqrt{5}$. The point is that $\frac{1}{2}(1+\sqrt{5}) $ is integral over $R$ but does not belong to $R$: a local integral domain of Krull dimension 1 is a DVR iff it is integrally closed. | |
| Aug 20, 2014 at 2:05 | comment | added | Neil Epstein | There are lots of examples. For instance, let $A={\mathbb Z}[\sqrt{-5}]$, $P = (2, 1+\sqrt{-5})$, $R=A_P$, and ${\mathfrak m} = P A_P$. Then $\mathfrak m$ is finitely generated but isn't principal, so $R$ isn't a valuation ring. As for a criterion, I believe that a local subring of a number field is a valuation ring iff its maximal ideal is principal. | |
| Aug 20, 2014 at 0:57 | history | asked | abcdxyz | CC BY-SA 3.0 |