You are here
Homeformally real field
Primary tabs
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 nonzero is not formally real; it is not even 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 $\tilde{\mathbb{Q}}\neq\mathbb{R}$.
Mathematics Subject Classification
12D15 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
Corrections
typo by pahio ✓
third root by pahio ✓
second condition by yark ✓
Real Closure of $\mathbb{Q}$ by mehdighasemi ✓
Comments
formally real fields are fields that are real rings
Maybe you should add an alternative "definition" or a remark.
A formally real field is just a field that is a real ring.
And that formally real fields are also known as real fields... I will add to my real rings a sentence that this ring is also called a "formally real ring".
The usage of the word "formally" was done earlier (the earliest excerpt I have seen was from a paper I saw from Tarski and one from Erdos.. or maybe both of them had written the same paper.. I have a bad memory). And then sometime later it just got too tedious to write formally real fields, so people dropped the "formally" part. But still a lot of mathematician do write "formally".