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
- 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.
References
- Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966
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