Inmathematics, adiscrete valuation is anintegervaluation on afieldK; that is, afunction:^[1]

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

satisfying the conditions:

${\displaystyle \nu (x\cdot y)=\nu (x)+\nu (y)}$

${\displaystyle \nu (x+y)\ge min\{\nu (x),\nu (y)\}}$

${\displaystyle \nu (x)=\mathrm{\infty }⟺x=0}$

for all${\displaystyle x,y\in K}$.

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

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

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

${\displaystyle {\mathcal{O}}_K:=\left\{x\in K\mid \nu (x)\ge 0\right\}}$

of${\displaystyle K}$, which is adiscrete valuation ring. Conversely, the valuation${\displaystyle \nu :A\to \mathbb{Z}\cup \{\mathrm{\infty }\}}$ on a discrete valuation ring${\displaystyle A}$ can be extended in a unique way to a discrete valuation on thequotient field${\displaystyle K=\text{Quot}(A)}$; the associated discrete valuation ring${\displaystyle {\mathcal{O}}_K}$ is just${\displaystyle A}$.

• For a fixedprime${\displaystyle p}$ and for any element${\displaystyle x\in \mathbb{Q}}$ different from zero write${\displaystyle x=p^j\frac{a}{b}}$ with${\displaystyle j,a,b\in \mathbb{Z}}$ such that${\displaystyle p}$ does not divide${\displaystyle a,b}$. Then${\displaystyle \nu (x)=j}$ is a discrete valuation on${\displaystyle \mathbb{Q}}$, called thep-adic valuation.
• Given aRiemann surface${\displaystyle X}$, we can consider the field${\displaystyle K=M(X)}$ ofmeromorphic functions${\displaystyle X\to \mathbb{C}\cup \{\mathrm{\infty }\}}$. For a fixed point${\displaystyle p\in X}$, we define a discrete valuation on${\displaystyle K}$ as follows:${\displaystyle \nu (f)=j}$ if and only if${\displaystyle j}$ is the largest integer such that the function${\displaystyle f(z)/(z-p{)}^j}$ can be extended to aholomorphic function at${\displaystyle p}$. This means: if${\displaystyle \nu (f)=j>0}$ then${\displaystyle f}$ has a root of order${\displaystyle j}$ at the point${\displaystyle p}$; if then${\displaystyle f}$ has apole of order${\displaystyle -j}$ at${\displaystyle p}$. In a similar manner, one also defines a discrete valuation on thefunction field of analgebraic curve for every regular point${\displaystyle p}$ on the curve.

More examples can be found in the article ondiscrete valuation rings.

• Commutative algebra
• Field (mathematics)