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.