Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function: ^[1]

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

satisfying the conditions:

${\displaystyle \nu (x\cdot y)=\nu (x)+\nu (y)}$

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

${\displaystyle \nu (x)=\infty \iff x=0}$

for all ${\displaystyle x,y\in K}$.

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

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

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

Examples

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

More examples can be found in the article on discrete valuation rings.

Citations

1. Cassels & Fröhlich 1967, p. 2.

References

• Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967), Algebraic Number Theory, Academic Press, Zbl   0153.07403
• Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN   978-0-8218-3259-2, MR   1915966