Euclidean field
For algebraic number fields whose ring of integers has a Euclidean algorithm, see Euclidean domain.
In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some y in K.
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Properties
- Every Euclidean field is an ordered Pythagorean field, but the converse is not true.
Examples
- The real numbers R with the usual operations and ordering form a Euclidean field.
- The field of real algebraic numbers
is a Euclidean field. - The field of hyperreal numbers is an Euclidean field.
Counterexamples
- The rational numbers Q with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in Q since the square root of 2 is irrational.
- The complex numbers C do not form a Euclidean field since they cannot be given the structure of an ordered field.