Euclidean ordered field
In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in.
The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Euclidean field contains it as an ordered subfield. In other words, the constructible numbers form the Euclidean closure of the rational numbers.
Properties
- Every Euclidean field is an ordered Pythagorean field, but the converse is not true.
- If E/''F is a finite extension, and E'' is Euclidean, then so is F. This "going-down theorem" is a consequence of the Diller–Dress theorem.
Examples
- The real constructible numbers, those lengths which can be constructed from a rational segment by ruler and compass constructions, form a Euclidean field.
- The real numbers 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 a Euclidean field.
Counterexamples
- The rational numbers with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in since the square root of 2 is irrational. By the going-down result above, no algebraic number field can be Euclidean.
- The complex numbers do not form a Euclidean field since they cannot be given the structure of an ordered field.
Euclidean closure