discrete valuation

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

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

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

is a subring of $K$ with sole maximal ideal

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 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 ), and sends the element $0\in K$ to $\mathrm{\infty }$. It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.