Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Discrete valuation
Summary
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.
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.
More examples can be found in the article on discrete valuation rings.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.