An ordered field $F$ is Euclidean if every non-negative element $a$ ($a\geq0$ is a square in $F$ (there exists $b\in F$ such that $b^2=a$ .

Examples

• $\mathbb{R}$ is Euclidean. -7-JG is not Euclidean because $2$ is not a square in $\mathbb{Q}$ (i.e., $\pm\sqrt{2}\notin \mathbb{Q}$ .
• $\mathbb{C}$ is not a Euclidean field because $\mathbb{C}$ is not an ordered field.
• The field of real constructible numbers is Euclidean.

A Euclidean field is an ordered Pythagorean field.

There are ordered fields that are Pythagorean but not Euclidean.