Euclidean field
An ordered field F is Euclidean if every non-negative element a (a≥0) is a square in F (there exists b∈F such that b2=a).
1 Examples
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ℝ is Euclidean.
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ℚ is not Euclidean because 2 is not a square in ℚ (i.e. (http://planetmath.org/Ie), ±√2∉ℚ).
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ℂ is not a Euclidean field because ℂ is not an ordered field (http://planetmath.org/MathbbCIsNotAnOrderedField).
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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 |
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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 |