# Formally real fields

## Definition

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

• $k$ is a formally real ring.

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

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

• The Witt group? of $k$ 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.