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.
$\mathrm{ord}(x+y)\geqq \mathrm{min}\{\mathrm{ord}x,\mathrm{ord}y\}$.
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 nonzero 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.
References
 1 E. Artin: Theory of Algebraic Numbers^{}. Lecture notes. Mathematisches Institut, Göttingen (1959).
 2 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
Title  order valuation 
Canonical name  OrderValuation 
Date of creation  20130322 16:53:28 
Last modified on  20130322 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 