Discrete valuation


In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function
satisfying the conditions
for all.
Note that often the trivial valuation which takes on only the values 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 we can associate the subring
of, which is a discrete valuation ring. Conversely, the valuation on a discrete valuation ring can be extended in a unique way to a discrete valuation on the quotient field ; the associated discrete valuation ring is just.

Examples

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