The “” of is reversed and the operation of is denoted by “”. The element 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.
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 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 .
then the function , defined in , clearly satisfies the above postulates of the order valuation.
In , 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.
- 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).
|Date of creation||2013-03-22 16:53:28|
|Last modified on||2013-03-22 16:53:28|
|Last modified by||pahio (2872)|
|Defines||exponent of field|
|Defines||zero of an element|
|Defines||pole of an element|