Discrete valuation explained

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

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

satisfying the conditions:

\nu(x ⋅ y) = \nu(x) + \nu(y) \nu(x + y)\geq min \ {\nu(x),\nu(y) \ } \nu(x) =inf ty\iff x =0

for all

x,y\in K .

Note that often the trivial valuation which takes on only the values

0 , inf ty 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 subringl{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 → \Z\cup\{ inf ty\} on a discrete valuation ring A can be extended in a unique way to a discrete valuation on the quotient field K = Quot (A) ; the associated discrete valuation ring l{O} _{K} is just A .

Examples

p and for any element x \in Q different from zero write x = p ^{j (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.X , we can consider the field K = M(X) of meromorphic functions X\to\Complex\cup\{ inf in\} . 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.