Top Qs
Timeline
Chat
Perspective

Formally real field

Field that can be equipped with an ordering From Wikipedia, the free encyclopedia

Remove ads

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

Alternative definitions

Summarize
Perspective

The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as (infinitely many) first-order 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:[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 1s.) This can be expressed in first-order logic by , , 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 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 PF. One uses this positive cone to define an ordering: ab if and only if ba belongs to P.

Remove ads

Real closed fields

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] and the non-negative elements are exactly the squares.

Notes

References

Loading content...
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads