Formally real field

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In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be extended with a (not necessarily unique) ordering which makes it an ordered field.

Alternative Definitions[edit]

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,[citation needed] 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][2]

  • −1 is not a sum of squares in F. In other words, the Stufe of F is infinite. (In particular, such a field must have characteristic 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 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[edit]

A formally real field with no formally real proper algebraic extension is a real closed field.[3] If K is formally real and Ω is an algebraically closed field containing K, then there is a real closed subfield of Ω containing K. A real closed field can be ordered in a unique way.[3]

Notes[edit]

  1. ^ Rajwade, Theorem 15.1.
  2. ^ Milnor and Husemoller (1973) p.60
  3. ^ a b Rajwade (1993) p.216

References[edit]