(more info)

Math for the people, by the people.

Main Menu

Euclidean field

(Definition)

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

Examples

$\mathbb{R}$ is Euclidean.
$\mathbb{Q}$ is not Euclidean because 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.

"Euclidean field" is owned by CWoo. [ full author list (3) ]

(view preamble | get metadata)

Also defines:

Euclidean

Log in to rate this entry.
(view current ratings)

Cross-references: Pythagorean field, square, ordered field
There are 13 references to this entry.

This is version 31 of Euclidean field, born on 2004-05-21, modified 2008-03-12.
Object id is 5869, canonical name is EuclideanField.
Accessed 4354 times total.

Classification:

12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact