1

Is there an example of a complete valuation ring $(R, m)$ which is not a DVR but such that $R/m$ is finite? Examples of valuation rings I have in mind are ring of integers of finite extensions of $\Bbb Q_p$, but those are DVR, or $\Bbb C_p$ but the residual field is infinite...

Thank you very much!

Orat
  • 4,065
Alphonse
  • 6,342
  • 1
  • 19
  • 48
  • 1
    The valuation ring of the field of Hahn series $\Bbb F_q[[T^{\Bbb Q}]]$ should do the trick. – Alphonse Oct 07 '19 at 19:33
  • 1
    In general there's a recipe for constructing a valuation with any prescribed value group and residue field. See my answer in https://math.stackexchange.com/questions/3415695/valuation-rings-only-have-two-prime-ideals/3581762#3581762. Using the notation of the above answer, start with any finite $\mathtt{k}$ and arbitrary $\Gamma$,.where $\Gamma$ is not isomorphic to $\mathbb{Z}$. Since completion is immediate, the completion of $\mathtt{k} ((t^{\Gamma}))$ should work. – Arpan Dutta Mar 15 '20 at 17:34

0 Answers0