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.

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
  • Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs. 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. 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.

External links[edit]