Formally real field

This article defines a field property: a property that can be evaluated to true/false for any field.
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Definition

Algebraic definition

A formally real field is a field in which $-1$ is not a sum of squares.

Definition in terms of total ordering

A formally real field is a field for which there exists a total ordering $\le$ under which the field becomes an ordered field. In other words, $\le$ satisfies the following conditions:

• $a\le b$ and $c\le d$ implies $a+c\le b+d$.
• $0\le 1$.
• $a\le b$ implies $-b\le -a$.
• $0\le a$ and $b\le c$ implies $ab\le ac$.

Relation with other properties

Stronger properties

• Real-closed field
• Archimedean field
• Euclidean field

Weaker properties

• Field of characteristic zero: For proof of the implication, refer Formally real implies characteristic zero and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic zero not implies formally real.