Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field "K"; that is, a function
satisfying the conditions
for all formula_5.
Note that often the trivial valuation which takes on only the values formula_6 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 formula_7 with discrete valuation formula_8 we can associate the subring
of formula_7, which is a discrete valuation ring. Conversely, the valuation formula_11 on a discrete valuation ring formula_12 can be extended in a unique way to a discrete valuation on the quotient field formula_13; the associated discrete valuation ring formula_14 is just formula_12.
Examples.
More examples can be found in the article on discrete valuation rings.