formally real field

A field $F$ is called formally real if $-1$ can not be expressed as a sum of squares (of elements of $F$ ).

Given a field $F$ , let $S_{F}$ be the set of all sums of squares in $F$ . The following are equivalent conditions that $F$ is formally real:

1.

$-1\notin S_{F}$
2.

$S_{F}\not=F$ and $\operatorname{char}(F)\neq 2$
3.

$\sum{a_{i}}^{2}=0$ implies each $a_{i}=0$ , where $a_{i}\in F$
4.

$F$ can be ordered (There is a total order $<$ which makes $F$ into an ordered field)

Some Examples:

•

$\mathbb{R}$ and $\mathbb{Q}$ are both formally real fields.
•

If $F$ is formally real, so is $F(\alpha)$ , where $\alpha$ is a root of an irreducible polynomial of odd degree in $F[x]$ . 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 $-1=i^{2}$ .
•

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 $k$ , a formally real field $K$ is said to be a real closure of $k$ if $K$ is an algebraic extension of $k$ and is real closed.

In the example above, $\mathbb{R}$ is real closed, and $\mathbb{Q}$ is not, whose real closure is $\tilde{\mathbb{Q}}$ . Furthermore, it can be shown that the real closure of a countable formally real field is countable, so that $\tilde{\mathbb{Q}}\neq\mathbb{R}$ .

Title	formally real field
Canonical name	FormallyRealField
Date of creation	2013-03-22 14:22:22
Last modified on	2013-03-22 14:22:22
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	20
Author	CWoo (3771)
Entry type	Definition
Classification	msc 12D15
Related topic	PositiveCone
Related topic	RealRing
Defines	formally real
Defines	real closed
Defines	real closure