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formally real field

Formally real fields

Definition

A field kk is formally real if one of the following equivalent conditions is satisfied:

  • kk is a formally real ring.

  • 1-1 is not a sum of squares in kk (a special case of the previous condition).

  • There exists an linear order on kk which makes kk into an ordered field.

  • The Witt group? of kk is not torsion.

Due to the last property, formally real fields play a special rôle in the theory of Witt groups? and in related fields of study such as motivic homotopy theory.

A formally real field which is the only formally real algebraic extension of itself is a real closed field; such a field admits a unique ordering.

Examples

category: algebra

Last revised on October 27, 2013 at 16:52:26. See the history of this page for a list of all contributions to it.