PlanetMath: formally real field

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formally real field

(Definition)

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:

$-1\notin S_F$
$S_F\not= F$ and $\operatorname{char}(F)\ne 2$
$\sum {a_i}^2=0$ implies each , where $a_i\in F$
can be ordered (There is a total order which makes into an ordered field)

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.

"formally real field" is owned by CWoo. [ full author list (2) ]

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See Also: positive cone

Also defines:

formally real

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Cross-references: even, characteristic, root of unity, degree, odd, irreducible polynomial, root, ordered field, total order, implies, the following are equivalent, squares, sum, field
There are 4 references to this entry.

This is version 13 of formally real field, born on 2004-05-18, modified 2007-07-07.
Object id is 5863, canonical name is FormallyRealField.
Accessed 2841 times total.

Classification:

12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )

Pending Errata and Addenda

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