is not an ordered field
is not an ordered field.
First, the following theorem will be proven:
is not an ordered ring.
Many facts that are used here are proven in the entry regarding basic facts about ordered rings.
Suppose that is an ordered ring under some total ordering . Note that and
Note also that . Thus, either or . In either case, , a contradiction.
It follows that is not an ordered ring. ∎
Because of theorem 2, no ring containing can be an ordered ring. It follows that is not an ordered field.
|Title||is not an ordered field|
|Date of creation||2013-03-22 16:17:25|
|Last modified on||2013-03-22 16:17:25|
|Last modified by||Wkbj79 (1863)|