Let $K$ be a local field, that is, complete with discrete value, and its residue field is finite. Then, is the integer ring of $K$ a DVR?
For example, p-adic number field's integer ring is p-adic integer ring, which is DVR. What about in general case ?
Thank you in advance.
Best Answer
Given an arbitrary field $K$ together with a non-trivial valuation $v:K\to \Bbb Z\cup\{\infty\}$ the valuation ring ${\cal O}_K=\{x\in K\mid v(x)\geq0\}$ is always a DVR, in particular if $K$ is a local field.
Note first that $M=\{x\in K\mid v(x)>0\}$ is the unique maximal ideal of ${\cal O}_K$ because ${\cal O}_K\setminus M=\{x\in K\mid v(x)=0\}={\cal O}_K^\times$. Thus ${\cal O}_K$ is a local ring. As $v$ is non-trivial ${\cal O}_K$ is not a field.
Let $I\subseteq {\cal O}_K$ be a non-zero ideal. Take $x\in I$ with minimal valuation $v(x)$. If $y$ is any element in $I$ we have $v(x^{-1}y)\geq0$, hence $y\in (x)$, thus $I=(x)$, i.e. ${\cal O}_K$ is a PID and therefore a DVR.
(This imitates the proof that a euclidean domain is a PID because the valuation restricted to ${\cal O}_K\setminus\{0\}$ gives a euclidean function on the valuation ring.)