# discrete valuation

A on a field $K$ is a valuation $|\cdot|:K\to\mathbb{R}$ whose image is a discrete subset of $\mathbb{R}$.

For any field $K$ with a discrete valuation $|\cdot|$, the set

 $R:=\{x\in K:|x|\leq 1\}$

is a subring of $K$ with sole maximal ideal

 $M:=\{x\in K:|x|<1\},$

and hence $R$ is a discrete valuation ring. Conversely, given any discrete valuation ring $R$, the field of fractions $K$ of $R$ admits a discrete valuation sending each element $x\in R$ to $c^{n}$, where $0 is some arbitrary fixed constant and $n$ is the order of $x$, and extending multiplicatively to $K$.

Note: Discrete valuations are often written additively instead of multiplicatively; under this alternate viewpoint, the element $x$ maps to $\log_{c}|x|$ (in the above notation) instead of just $|x|$. This transformation reverses the order of the absolute values (since $c<1$), and sends the element $0\in K$ to $\infty$. It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.

Title discrete valuation DiscreteValuation 2013-03-22 13:59:14 2013-03-22 13:59:14 djao (24) djao (24) 6 djao (24) Definition msc 13F30 msc 12J20 rank one valuations rank-one valuations DiscreteValuationRing Valuation