Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function

[math]\displaystyle{ \nu:K\to\mathbb Z\cup\{\infty\} }[/math]

satisfying the conditions

[math]\displaystyle{ \nu(x\cdot y)=\nu(x)+\nu(y) }[/math]

[math]\displaystyle{ \nu(x+y)\geq\min\big\{\nu(x),\nu(y)\big\} }[/math]

[math]\displaystyle{ \nu(x)=\infty\iff x=0 }[/math]

for all [math]\displaystyle{ x,y\in K }[/math].

Note that often the trivial valuation which takes on only the values [math]\displaystyle{ 0,\infty }[/math] 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 [math]\displaystyle{ K }[/math] with discrete valuation [math]\displaystyle{ \nu }[/math] we can associate the subring

[math]\displaystyle{ \mathcal{O}_K := \left\{ x \in K \mid \nu(x) \geq 0 \right\} }[/math]

of [math]\displaystyle{ K }[/math], which is a discrete valuation ring. Conversely, the valuation [math]\displaystyle{ \nu: A \rightarrow \Z\cup\{\infty\} }[/math] on a discrete valuation ring [math]\displaystyle{ A }[/math] can be extended in a unique way to a discrete valuation on the quotient field [math]\displaystyle{ K=\text{Quot}(A) }[/math]; the associated discrete valuation ring [math]\displaystyle{ \mathcal{O}_K }[/math] is just [math]\displaystyle{ A }[/math].

Examples

• For a fixed prime [math]\displaystyle{ p }[/math] and for any element [math]\displaystyle{ x \in \mathbb{Q} }[/math] different from zero write [math]\displaystyle{ x = p^j\frac{a}{b} }[/math] with [math]\displaystyle{ j, a,b \in \Z }[/math] such that [math]\displaystyle{ p }[/math] does not divide [math]\displaystyle{ a,b }[/math], then [math]\displaystyle{ \nu(x) = - j }[/math] is a discrete valuation on [math]\displaystyle{ \Q }[/math], called the p-adic valuation.
• Given a Riemann surface [math]\displaystyle{ X }[/math], we can consider the field [math]\displaystyle{ K=M(X) }[/math] of meromorphic functions [math]\displaystyle{ X\to\Complex\cup\{\infin\} }[/math]. For a fixed point [math]\displaystyle{ p\in X }[/math], we define a discrete valuation on [math]\displaystyle{ K }[/math] as follows: [math]\displaystyle{ \nu(f)=j }[/math] if and only if [math]\displaystyle{ j }[/math] is the largest integer such that the function [math]\displaystyle{ f(z)/(z-p)^j }[/math] can be extended to a holomorphic function at [math]\displaystyle{ p }[/math]. This means: if [math]\displaystyle{ \nu(f)=j\gt 0 }[/math] then [math]\displaystyle{ f }[/math] has a root of order [math]\displaystyle{ j }[/math] at the point [math]\displaystyle{ p }[/math]; if [math]\displaystyle{ \nu(f)=j\lt 0 }[/math] then [math]\displaystyle{ f }[/math] has a pole of order [math]\displaystyle{ -j }[/math] at [math]\displaystyle{ p }[/math]. In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point [math]\displaystyle{ p }[/math] 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