formally real field
A field is called formally real if can not be expressed as a sum of squares (of elements of ).
Given a field , let be the set of all sums of squares in . The following are equivalent conditions that is formally real:
implies each , where
and are both formally real fields.
If is formally real, so is , where is a root of an irreducible polynomial of odd degree in . As an example, is formally real, where is a third root of unity.
is not formally real since .
Any field of characteristic non-zero is not formally real; it is not orderable.
A formally real field is said to be real closed if any of its algebraic extension which is also formally real is itself. For any formally real field , a formally real field is said to be a real closure of if is an algebraic extension of and is real closed.
In the example above, is real closed, and is not, whose real closure is . Furthermore, it can be shown that the real closure of a countable formally real field is countable, so that .
|Title||formally real field|
|Date of creation||2013-03-22 14:22:22|
|Last modified on||2013-03-22 14:22:22|
|Last modified by||CWoo (3771)|