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 $$K$$ 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 $$\nu: A \rightarrow \Z\cup\{\infty\}$$ 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 $$j, a,b \in \Z$$ 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 $$X\to\Complex\cup\{\infin\}$$. 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.