# 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.

$-1\notin S_{F}$

2. 2.

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

3. 3.

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

4. 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 FormallyRealField 2013-03-22 14:22:22 2013-03-22 14:22:22 CWoo (3771) CWoo (3771) 20 CWoo (3771) Definition msc 12D15 PositiveCone RealRing formally real real closed real closure