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. $S_F\ne F$ and $\mathrm{char}(F)\ne 2$

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:

• $ℝ$ 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 $\omega \ne 1$ is a third root of unity.

• $ℂ$ 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, $ℝ$ 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 $\tilde{\mathbb{Q}}\ne ℝ$.