Affiliation:
1. Institute of Informatics , University of Gdańsk , Poland
Abstract
Summary
In this article we extend the algebraic theory of ordered fields [6], [8] in Mizar. We introduce extensions of orderings: if E is a field extension of F, then an ordering P of F extends to E, if there exists an ordering O of E containing P. We first prove some necessary and su cient conditions for P being extendable to E, in particular that P extends to E if and only if the set
Q
S
E
:
=
{
∑
a
*
b
2
|
a
∈
P
,
b
∈
E
}
QS\,\,E: = \left\{ {\sum {a*{b^2}|a \in P,\,\,b \in E} } \right\}
is a preordering of E – or equivalently if and only if −1 / ∉ QS E. Then we show for non-square a ∈ F that P extends to
F
(
a
)
F\left( {\sqrt a } \right)
if and only if P and finally that every ordering P of F extends to E if the degree of E over F is odd.