Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function:^[1]

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

satisfying the conditions:

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

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

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

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

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.