order valuation

Given a Krull valuation $|.|$ of a field $K$ as a mapping of $K$ to an ordered group $G$ (with operation “$\cdot$”) equipped with $0$, one may use for the an alternative notation “ord”:

The “” of $G$ is reversed and the operation of $G$ is denoted by “$+$”.  The element $0$ of $G$ is denoted as $\mathrm{\infty }$, thus $\mathrm{\infty }$ is greater than any other element of $G$.  When we still call the valuation the order of $K$ and instead of $|x|$ write  $\mathrm{ord}x$, the valuation postulates read as follows.

1. $\mathrm{ord}x=\mathrm{\infty }$   iff   $x=0$;

2. $\mathrm{ord}xy=\mathrm{ord}x+\mathrm{ord}y$;

3. $\mathrm{ord}(x+y)\geqq \min \{\mathrm{ord}x,\mathrm{ord}y\}$.

We must emphasize that the order valuation is nothing else than a Krull valuation.  The name order comes from complex analysis, where the “places” zero (http://planetmath.org/ZeroOfAFunction) and pole (http://planetmath.org/Pole) of a meromorphic function with their orders have a fully analogous meaning as have the corresponding concepts place (http://planetmath.org/PlaceOfField) and order valuation in the valuation theory.  Thus also a place $\phi$ of a field is called a zero of an element $a$ of the field, if  $\phi (a)=0$,  and a pole of an element $b$ of the field, if  $\phi (b)=\mathrm{\infty }$;  then e.g. the equation  $\phi (a)=0$  implies always that  $\mathrm{ord}a>0$.

Example.  Let $p$ be a given positive prime number.  Any non-zero rational number $x$ can be uniquely expressed in the form

$$x=p^{n}u,$$

in which $n$ is an integer and the rational number $u$ is by $p$ indivisible, i.e. when reduced to lowest terms, $p$ divides neither its numerator nor its denominator.  If we define

${\mathrm{ord}}_{p}x=\{\begin{array}{cc}\mathrm{\infty }\mathrm{for}x=0,\hfill & \\ n\mathrm{for}x=p^{n}u\ne 0,\hfill & \end{array}$

then the function ${\mathrm{ord}}_p$, defined in $\mathbb{Q}$, clearly satisfies the above postulates of the order valuation.

In [2], an order valuation having only integer values is called the exponent of the field (der Exponent des Körpers); this name apparently motivated by the exponent $n$ of $p$.  Such an order valuation is a special case of the discrete valuation.  Note, that an arbitrary order valuation need not be a discrete valuation, since the values need not be real numbers.

Title	order valuation
Canonical name	OrderValuation
Date of creation	2013-03-22 16:53:28
Last modified on	2013-03-22 16:53:28
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	19
Author	pahio (2872)
Entry type	Definition
Classification	msc 13F30
Classification	msc 13A18
Classification	msc 12J20
Classification	msc 11R99
Synonym	additive valuation
Related topic	KrullValuation
Related topic	Valuation
Related topic	PAdicValuation
Related topic	DiscreteValuation
Related topic	ZerosAndPolesOfRationalFunction
Related topic	AlternativeDefinitionOfValuation2
Related topic	StrictDivisibility
Related topic	ExponentValuation2
Related topic	DivisibilityOfPrimePowerBinomialCoefficients
Defines	exponent of field
Defines	zero
Defines	zero of an element
Defines	pole
Defines	pole of an element