Euclidean field

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 numbers R with the usual operations and ordering form a Euclidean field.
• The field of real algebraic numbers is a Euclidean field.
• The real constructible numbers, those lengths which can be constructed from a rational segment by ruler and compass constructions, form a Euclidean field.
• The field of hyperreal numbers is a 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. By the going-down result above, no algebraic number field can be Euclidean.
• The complex numbers C do not form a Euclidean field since they cannot be given the structure of an ordered field.

  Euclidean closure