# 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 $\infty$, thus $\infty$ 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. 1.

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

2. 2.

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

3. 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 $\varphi$ of a field is called a zero of an element $a$ of the field, if  $\varphi(a)=0$,  and a pole of an element $b$ of the field, if  $\varphi(b)=\infty$;  then e.g. the equation$\varphi(a)=0$  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

 $\displaystyle\mathrm{ord}_{p}x\;=\;\begin{cases}\infty\,\,\,\mathrm{for}\,\,\,% x=0,\\ n\,\,\,\mathrm{for}\,\,\,x=p^{n}u\neq 0,\end{cases}$

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 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