Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function:[1] satisfying the conditions: for all

Note that often the trivial valuation which takes on only the values

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

, which is a discrete valuation ring.

can be extended in a unique way to a discrete valuation on the quotient field

; the associated discrete valuation ring

More examples can be found in the article on discrete valuation rings.