formally real field

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 SF be the set of all sums of squares in F. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath conditions that F is formally real:

  1. 1.


  2. 2.

    SFF and char(F)2

  3. 3.

    ai2=0 implies each ai=0, where aiF

  4. 4.

    F can be ordered (There is a total orderMathworldPlanetmath < which makes F into an ordered field)

Some Examples:

  • and are both formally real fields.

  • If F is formally real, so is F(α), where α is a root of an irreducible polynomial of odd degree in F[x]. As an example, (23ω) is formally real, where ω1 is a third root of unity.

  • is not formally real since -1=i2.

  • Any field of characteristic non-zero is not formally real; it is not orderable.

A formally real field is said to be real closed if any of its algebraic extensionMathworldPlanetmath which is also formally real is itself. For any formally real field k, a formally real field K is said to be a real closure of k if K is an algebraic extension of k and is real closed.

In the example above, is real closed, and is not, whose real closure is ~. Furthermore, it can be shown that the real closure of a countableMathworldPlanetmath formally real field is countable, so that ~.

Title formally real field
Canonical name FormallyRealField
Date of creation 2013-03-22 14:22:22
Last modified on 2013-03-22 14:22:22
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 20
Author CWoo (3771)
Entry type Definition
Classification msc 12D15
Related topic PositiveCone
Related topic RealRing
Defines formally real
Defines real closed
Defines real closure