Editing Formally real field

{{Short description|Field that can be equipped with an ordering}}
{{More citations needed|date=December 2009}}
In [[mathematics]], in particular in [[field theory (mathematics)|field theory]] and [[Real algebraic geometry|real algebra]], a '''formally real field''' is a [[Field (mathematics)|field]] that can be equipped with a (not necessarily unique) ordering that makes it an [[ordered field]].

==Alternative definitions==
The definition given above is not a [[First-order logic|first-order]] definition, as it requires quantifiers over [[Set (mathematics)|sets]]. However, the following criteria can be coded as (infinitely many) first-order [[Sentence (mathematical logic)|sentences]] in the language of fields and are equivalent to the above definition.

A formally real field ''F'' is a field that also satisfies one of the following equivalent properties:<ref>Rajwade, Theorem 15.1.</ref><ref>Milnor and Husemoller (1973) p.60</ref>

* −1 is not a sum of [[Square number|square]]s in ''F''.  In other words, the [[Stufe (algebra)|Stufe]] of ''F'' is infinite. (In particular, such a field must have [[Characteristic (algebra)|characteristic]] 0, since in a field of characteristic ''p'' the element −1 is a sum of 1s.)  This can be expressed in first-order logic by <math>\forall x_1 (-1 \ne x_1^2)</math>, <math>\forall x_1 x_2 (-1 \ne x_1^2 + x_2^2)</math>, etc., with one sentence for each number of variables.
* There exists an element of ''F'' that is not a sum of squares in ''F'', and the characteristic of ''F'' is not 2.
* If any sum of squares of elements of ''F'' equals zero, then each of those elements must be zero.

It is easy to see that these three properties are equivalent. It is also easy to see that a field that admits an ordering must satisfy these three properties.

A proof that if ''F'' satisfies these three properties, then ''F'' admits an ordering uses the notion of [[Ordered field#Def 2: A positive cone of F|prepositive cones]] and positive cones. Suppose −1 is not a sum of squares; then a [[Zorn's Lemma]] argument shows that the prepositive cone of sums of squares can be extended to a positive cone {{nowrap|''P'' ⊆ ''F''}}. One uses this positive cone to define an ordering: {{nowrap|''a'' ≤ ''b''}} if and only if {{nowrap|''b''&thinsp;−&thinsp;''a''}} belongs to ''P''.

==Real closed fields==

A formally real field with no formally real proper [[algebraic extension]] is a [[real closed field]].<ref name=R216>Rajwade (1993) p.216</ref>  If ''K'' is formally real and Ω is an [[algebraically closed field]] containing ''K'', then there is a real closed [[Field extension|subfield]] of Ω containing ''K''.  A real closed field can be ordered in a unique way,<ref name=R216/> and the non-negative elements are exactly the squares.

==Notes==
{{Reflist}}

==References==
*{{cite book | first1=John | last1=Milnor | authorlink=John Milnor | first2=Dale | last2=Husemoller | authorlink2=Dale Husemoller | title=Symmetric bilinear forms | publisher=Springer | year=1973 | isbn=3-540-06009-X }}
* {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}

{{DEFAULTSORT:Formally Real Field}}
[[Category:Field (mathematics)]]
[[Category:Ordered groups]]

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