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 .
For a fixed prime and for any element different from zero write with such that does not divide , then is a discrete valuation on , called the p-adic valuation.
Given a Riemann surface, we can consider the field of meromorphic functions. For a fixed point , we define a discrete valuation on as follows: if and only if is the largest integer such that the function can be extended to a holomorphic function at . This means: if then has a root of order at the point ; if then has a pole of order at . In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point on the curve.