( is additively closed)
( is multiplicatively closed)
As it turns out, a field endowed with a pre-positive cone has an order structure. The field is called a formally real (http://planetmath.org/FormallyRealField), orderable, or ordered field. Before defining what this “order” is, let’s do some preliminary work. Let be a pre-positive cone of a field . By Zorn’s Lemma, the set of pre-positive cones extending has a maximal element . It can be shown that has two additional properties:
First, suppose there is . Let . Then and so is strictly contained in . Clearly, and is easily seen to be additively closed. Also, is multiplicatively closed as the equation demonstrates. Since is a maximal and properly contains , is not a pre-positive cone, which means . Write . Then . Since , , , contradicting the assumption that . Therefore, .
For the second part, suppose . Since , . If , then , a contradiction. ∎
A subset of a field satisfying conditions 1, 2, 5 and 6 is called a positive cone of . A positive cone is a pre-positive cone. If , then either or . In either case, . Next, if , then . But , we have , contradicting Condition 6 of .
Now, define a binary relation , on by:
Thus, is a field ordered by .
Remark. Positive cones may be defined for more general ordered algebraic structures, such as partially ordered groups, or partially ordered rings.
- 1 A. Prestel, Lectures on Formally Real Fields, Springer, 1984
|Date of creation||2013-03-22 14:46:54|
|Last modified on||2013-03-22 14:46:54|
|Last modified by||CWoo (3771)|