In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular in field theory

Field theory (mathematics)

Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....

 and real algebra

Real algebraic geometry

In mathematics, real algebraic geometry is the study of real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them ....

, a formally real field is a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 that admits an ordering which makes it an ordered field

Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...

.

Alternative Definitions

The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as first-order sentences in the language of fields, and are equivalent to the above definition.

A formally real field F is a field that satisfies in addition one of the following equivalent properties:

• −1 is not a sum of square
  Square number
  In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
  
  s in F. (In particular, such a field must have characteristic
  Characteristic (algebra)
  In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
  
   0, since in a field of characteristic p the element −1 is a sum of 1's.)
• There exists an element of F which 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 which 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 prepositive cones and positive cones. Suppose −1 is not a sum of squares, then a Zorn's Lemma

Zorn's lemma

Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...

 argument shows that the prepositive cone of sums of squares can be extended to a positive cone P⊂F. One uses this positive cone to define an ordering: a≤b if and only if b-a belongs to P.

Real Closed Fields

A formally real field with no formally real algebraic extension is a real closed field

Real closed field

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...

.