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).
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.
Last revised on August 15, 2020 at 04:55:12. See the history of this page for a list of all contributions to it.