You are here
HomeEuclidean field
Primary tabs
Euclidean field
An ordered field $F$ is Euclidean if every nonnegative 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., $\pm\sqrt{2}\notin\mathbb{Q}$).

$\mathbb{C}$ 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.
Defines:
Euclidean
Related:
ConstructibleNumbers, EuclideanNumberField
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
12D15 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
Oct 7
new question: Lorenz system by David Bankom
Oct 19
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag
Sep 26
new question: Latent variable by adam_reith
new question: Lorenz system by David Bankom
Oct 19
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag
Sep 26
new question: Latent variable by adam_reith