Discrete valuation

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

satisfying the conditions

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.

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

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