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ArticleOriginal scientific text

Title

Ordered fields and the ultrafilter theorem

Authors 1, 2, 1

Affiliations

  1. Fachbereich Mathematik, Universität Dortmund, D-44221 Dortmund, Germany
  2. CNRS-UFR de Mathématiques, Université Paris 7, 2 Place Jussieu, Case 7012, F-75251 Paris Cedex 05, France

Abstract

We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.

Bibliography

  1. E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg 5 (1927), 100-115.
  2. E. Artin, Schreier, O. Algebraische Konstruktion reeller Körper, ibid., 85-99.
  3. T. Jech, The Axiom of Choice, North-Holland, 1973.
  4. H. Lombardi and M.-F. Roy, Constructive elementary theory of ordered fields, in: Effective Methods in Algebraic Geometry, Progr. Math. 94, Birkhäuser, 1991, 249-262.
  5. T. Sander, Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma, J. Pure Appl. Algebra 73 (1991), 165-180.
  6. A. Tarski, Prime ideal theorems for set algebras and ordering principles, preliminary report, Bull. Amer. Math. Soc. 60 (1954), 391.
Fundamenta Mathematicae
1999/159/3
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Pages:
231-241
Main language of publication
English
Received
1998-03-03
Accepted
1998-05-05
Published
1999
Exact and natural sciences