Euclidean field

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

A Euclidean field is a field satisfying the following equivalent conditions:

1. The field is a formally real field (i.e., $-1$ is not a square) and the set of squares is a subgroup of index two in the multiplicative group of the field.
2. The field does not have characteristic two, and for every element $D$, exactly one of the elements $D$ and $-D$ is a square.

Examples

• Field of constructible real numbers
• Field of real algebraic numbers
• Field of real numbers

Relation with other properties

Stronger properties

• Real-closed field

Weaker properties

• Formally real field
• Pythagorean field
• Field of characteristic zero