Discrete valuation

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

\nu :K\to \mathbb {Z} \cup \{\infty \}

satisfying the conditions

\nu (x\cdot y)=\nu (x)+\nu (y)

\nu (x+y)\geq \min {\big \{}\nu (x),\nu (y){\big \}}

\nu (x)=\infty \iff x=0

for all $x,y\in K$ .

Note that often the trivial valuation which takes on only the values $0,\infty$ 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 $\nu$ we can associate the subring

{\mathcal {O}}_{K}:=\left\{x\in K\mid \nu (x)\geq 0\right\}

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

Examples

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