# Euclidean field

An ordered field $F$ is Euclidean if every non-negative element $a$ ($a\geq 0$) is a square in $F$ (there exists $b\in F$ such that $b^{2}=a$).

## 1 Examples

• $\mathbb{R}$ is Euclidean.

• $\mathbb{Q}$ is not Euclidean because $2$ is not a square in $\mathbb{Q}$ (i.e. (http://planetmath.org/Ie), $\pm\sqrt{2}\notin\mathbb{Q}$).

• $\mathbb{C}$ is not a Euclidean field because $\mathbb{C}$ is not an ordered field (http://planetmath.org/MathbbCIsNotAnOrderedField).

• The field of real constructible numbers (http://planetmath.org/ConstructibleNumbers) is Euclidean.

A Euclidean field is an ordered Pythagorean field.

There are ordered fields that are Pythagorean but not Euclidean.

Title Euclidean field EuclideanField 2013-03-22 14:22:39 2013-03-22 14:22:39 CWoo (3771) CWoo (3771) 34 CWoo (3771) Definition msc 12D15 ConstructibleNumbers EuclideanNumberField Euclidean