# Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field k, that is a function

${\displaystyle \nu :k\to \mathbb {Z} \cup \{\infty \}}$

satisfying the conditions

${\displaystyle \nu (x\cdot y)=\nu (x)+\nu (y)}$
${\displaystyle \nu (x+y)\geq \min {\big \{}\nu (x),\nu (y){\big \}}}$
${\displaystyle \nu (x)=\infty \iff x=0.}$

Note that often the trivial valuation which takes on only the values ${\displaystyle 0,\infty }$ is explicitly excluded.

A field with a non-trivial discrete valuation is called a discrete valuation field.

## Discrete valuation rings and valuations on fields

To every field with discrete valuation ${\displaystyle \nu }$ we can associate the subring

${\displaystyle {\mathcal {O}}_{k}:=\left\{x\in k\mid \nu (x)\geq 0\right\}}$

of ${\displaystyle k}$, which is a discrete valuation ring. Conversely, the valuation ${\displaystyle \nu :A\rightarrow \mathbb {Z} \cup \{\infty \}}$ on a discrete valuation ring ${\displaystyle A}$ can be extended to a valuation on the quotient field ${\displaystyle {\text{Quot}}(A)}$ giving a discrete valued field ${\displaystyle k}$, whose associated discrete valuation ring ${\displaystyle {\mathcal {O}}_{k}}$ is just ${\displaystyle A}$.

## References

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