Harman Patil (Editor)

Euclidean field

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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.

Contents

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 R ∩ Q ¯ is a Euclidean field.
  • The real constructible numbers, those (signed) 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

    The Euclidean closure of an ordered field K is an extension of K in the quadratic closure of K which is maximal with respect to being an ordered field with an order extending that of K.

    References

    Euclidean field Wikipedia
    (Text) CC BY-SA