Euclidean field

An ordered field F is Euclidean if every non-negative element a (a0) is a square in F (there exists bF such that b2=a).

1 Examples

  • is Euclidean.

  • is not Euclidean because 2 is not a square in (i.e. (, ±2).

  • is not a Euclidean field because is not an ordered field (

  • The field of real constructible numbers ( is Euclidean.

A Euclidean field is an ordered Pythagorean fieldMathworldPlanetmath.

There are ordered fields that are Pythagorean but not Euclidean.

Title Euclidean field
Canonical name EuclideanField
Date of creation 2013-03-22 14:22:39
Last modified on 2013-03-22 14:22:39
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 34
Author CWoo (3771)
Entry type Definition
Classification msc 12D15
Related topic ConstructibleNumbers
Related topic EuclideanNumberField
Defines Euclidean