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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:

Some Examples:

$\mathbb{R}$ and $\mathbb{Q}$ are both formally real fields.
If is formally real, so is $F(\alpha)$ , where $\alpha$ is a root of an irreducible polynomial of odd degree in . As an example, $\mathbb{Q}(\sqrt[3]{2}\omega)$ is formally real, where $\omega\not= 1$ is a third root of unity.
$\mathbb{C}$ is not formally real since .
Any field of characteristic non-zero is not formally real; it is not even orderable.

1	Total Order
1	Root
1	Field
1	Summation
1	Even Number
1	Implication
1	Ordered Ring
2	Square Of A Number
2	Root Of Unity
3	Irreducible Polynomial
3	Characteristic
3	Order And Degree Of Polynomial
3	T F A E
5	Odds Ratio