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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 $\tilde{\mathbb{Q}}\neq\mathbb{R}$.
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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".