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