Euclidean field

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.

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

Examples

Every real closed field is a Euclidean field. The following examples are also real closed fields.

  • 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.[4] By the going-down result above, no algebraic number field can be Euclidean.[2]
  • The complex numbers 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.[5] It is also the smallest subfield of the algebraic closure of K that is a Euclidean field and is an ordered extension of K.

References

  1. Martin (1998) p. 89
  2. Lam (2005) p.270
  3. Martin (1998) pp. 35–36
  4. Martin (1998) p. 35
  5. Efrat (2006) p. 177
  • Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs. Vol. 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002.
  • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
  • Martin, George E. (1998). Geometric Constructions. Undergraduate Texts in Mathematics. Springer-Verlag. ISBN 0-387-98276-0. Zbl 0890.51015.
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