Valuation ring

Last updated January 27, 2022

Let $K$ be a field and $|\cdot|:K\to\mathbb{Z}_{\geq 0}$ a nonarchimedean absolute value.

Definition. The valuation ring of $K$ consists of all elements with absolute value $\leq 1$: $$ O={x\in K:|x|\leq 1} $$

Let $U=O^\ast=O\setminus p$ (see properties), i.e. those elements with absolute value 1. Then $U$ is a subgroup of $K^\ast$ and $|\cdot|:K^\ast/U\to\mathbb{R}^\ast_{>0}$ is an inclusion.

Definition. $|\cdot|$ is called discrete when $|K^\ast|$ under the above map is a discrete subgroup of $\mathbb{R}^\ast_{>0}$, i.e. is trivial or infinite cyclic.

Properties

• The maximal ideal of $O$ is all elements with absolute value $<1$: $$p={x\in K\mid|x|<1}$$
  • $p\neq (0)$ if and only if $|\cdot|$ is nontrivial
• For every nonzero $x\in K$, either $x\in O$ or $x^{-1}\in O$ (possibly both).
• $O$ is local and $K=O_{(0)}$
• $O$ is integrally closed.
• $|\cdot|$ is discrete if and only if $p$ is a principal ideal.