## You are here

Homeorder valuation

## Primary tabs

# 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 valuation an alternative notation “ord”:

The order “$<$” 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. $\mathrm{ord}\,x\,=\,\infty$ 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 and pole of a meromorphic function with their orders have a fully analogous meaning as have the corresponding concepts place 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).

## Mathematics Subject Classification

13F30*no label found*13A18

*no label found*12J20

*no label found*11R99

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections