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\$

satisfying the conditions:

:$\nu(x\cdot y)=\nu(x)+\nu(y)$
:$\nu(x+y)\geq\min\big\$
:$\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 $K$ with discrete valuation $\nu$ we can associate the subring

::$\mathcal_K := \left\$

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

Examples

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

Citations

References

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{{DEFAULTSORT:Discrete Valuation
Commutative algebra
Field (mathematics)