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 subringof, 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
- 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.