Discrete valuation

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

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

satisfying the conditions:

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

$\nu (x+y)\ge \min \{\nu (x),\nu (y)\}$

$\nu (x)=\mathrm{\infty }⟺x=0$

for all $x,y\in K$.

Note that often the trivial valuation which takes on only the values $0,\mathrm{\infty }$ 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 $K$ with discrete valuation $\nu$ we can associate the subring

${{𝒪}}_{\nu }:=\{x\in K\mid \nu (x)\ge 0\}$

of $K$, which is a discrete valuation ring. Conversely, the valuation $\nu :A\to \mathbb{\mathbb{Z}}\cup \{\mathrm{\infty }\}$ 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 ${{𝒪}}_{\nu }$ is just $A$.

Discrete valuation rings ${{𝒪}}_{\nu }$ are local rings with maximal ideal

${\mathfrak{𝔪}}_{\nu }:=\{x\in {{𝒪}}_{\nu }\mid \nu (x)>0\},$

so there is a notion of residue field $\kappa ={{𝒪}}_{\nu }/{\mathfrak{𝔪}}_{\nu }$. Some authors call ${{𝒪}}_{\nu }$ a local $K$-algebra when $\kappa$ and $K$ are isomorphic.

Examples

• For a fixed prime $p$ and for any element $x\in \mathbb{\mathbb{Q}}$ different from zero write $x=p^j\frac{a}{b}$ with $j,a,b\in \mathbb{\mathbb{Z}}$ such that $p$ does not divide $a,b$. Then $\nu (x)=j$ is a discrete valuation on $\mathbb{\mathbb{Q}}$, called the p-adic valuation.
• Given a Riemann surface $X$, we can consider the field $K=M(X)$ of meromorphic functions $X\to \mathbb{ℂ}\cup \{\mathrm{\infty }\}$. 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

• Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967), Algebraic Number Theory, Academic Press 
• 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