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formally real field (Definition)

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 $\operatorname{char}(F)\ne 2$
  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:

  • $\mathbb{R}$ 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 even orderable.




"formally real field" is owned by CWoo. [ full author list (3) ]
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See Also: positive cone, real ring

Also defines:  formally real
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Cross-references: characteristic, root of unity, degree, odd, irreducible polynomial, root, ordered field, total order, implies, the following are equivalent, squares, sum, field
There are 4 references to this entry.

This is version 14 of formally real field, born on 2004-05-18, modified 2009-03-20.
Object id is 5863, canonical name is FormallyRealField.
Accessed 3726 times total.

Classification:
AMS MSC12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )

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formally real fields are fields that are real rings by jocaps on 2009-03-20 21:16:23
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".
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