nLab formally real field

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. (This requires Zorn's Lemma.)

• 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 (not requiring Zorn's Lemma).

Examples

• The field of real numbers is formally real, and even a real closed field.

• The field $\mathbb{Q}$ of rational numbers is formally real but not real closed.

• Finite fields (or more generally fields of nonzero characteristic) and algebraically closed fields (such as the fields of complex numbers and algebraic numbers) are never formally real.

category: algebra