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

The order “” of is reversed and the operation of is denoted by “”. The element 0 of is denoted as , thus is greater than any other element of . When we still call the valuation the order of and instead of write , the valuation postulates read as follows.

1. iff ;
2. ;
3. .

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 of a field is called a zero of an element of the field, if , and a pole of an element of the field, if ; then e.g. the equation implies always that .

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

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 of . 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.

Bibliography

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).