# Integer ring of local field is DVR?

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 ?

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.)