Timeline for Extension of a complete discrete valuation ring
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 4, 2015 at 12:06 | comment | added | KConrad | That example contains essentially all the seeds of the general statement. Do you know how to extend an absolute value from a complete field to any finite extension, and then further to the algebraic closure? Do you see anything about $K = \mathbf Q_p$ that doesn't extend to the general case? | |
| Apr 4, 2015 at 7:08 | comment | added | rime | First of all thank you for your suggestion, but sorry, maybe I didn't explain well my question. I know that this is a good example of an extension not producing a dvr and I agree that on $\mathbb{C}_{p}$ the valuation has rank $1$. But what I'm trying to prove is the general fact that an extension constructed in this way is again a complete rank $1$ valuation ring. | |
| Apr 3, 2015 at 22:43 | comment | added | KConrad | Being height $1$ is equivalent to the value group being a subgroup of $\mathbf R$. Since you say you're looking for an example, try $R = {\mathbf Z}_p$ and let the algebraic extension of $K = {\mathbf Q}_p$ be the algebraic closure of $K$, so $L = {\mathbf C}_p$. Do you agree that the valuation on $L$ extending that on $K$ has rank $1$? | |
| Apr 3, 2015 at 22:33 | review | First posts | |||
| Apr 3, 2015 at 23:11 | |||||
| Apr 3, 2015 at 22:33 | history | asked | rime | CC BY-SA 3.0 |