Euclidean field

From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about ordered fields. For algebraic number fields whose ring of integers has a Euclidean algorithm, see Norm-Euclidean field. For the class of models in statistical mechanics, see Euclidean field theory.

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.

Properties[edit]

Examples[edit]

Counterexamples[edit]

Euclidean closure[edit]

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]

References[edit]

  1. ^ Martin (1998) p. 89
  2. ^ a b Lam (2005) p.270
  3. ^ Martin (1998) pp. 35–36
  4. ^ Martin (1998) p. 35
  5. ^ Efrat (2006) p. 177

External links[edit]