The Boolean space of orderings of a field
HTML articles powered by AMS MathViewer
- by Thomas C. Craven PDF
- Trans. Amer. Math. Soc. 209 (1975), 225-235 Request permission
Abstract:
It has been pointed out by Knebusch, Rosenberg and Ware that the set $X$ of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings $W(a) = \{ < {\text { in }}X|a < 0\}$ the Harrison subbasis of $X$. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space $X$, there exists a formally real field $F$ such that $X$ is homeomorphic to the space of orderings on $F$. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.References
-
N. Bourbaki, Éléments de mathématique. XIV. Part. 1. Les structures fondamentales de l’analyse. Livre II: Algèbre. Chap. 6, 2nd ed., Actualités Sci. Indust., no. 1179, Hermann, Paris, 1964.
- Thomas C. Craven, The topological space of orderings of a rational function field, Duke Math. J. 41 (1974), 339–347. MR 349639
- Richard Elman and T. Y. Lam, Quadratic forms over formally real fields and pythagorean fields, Amer. J. Math. 94 (1972), 1155–1194. MR 314878, DOI 10.2307/2373568
- Richard Elman, Tsit Yuen Lam, and Alexander Prestel, On some Hasse principles over formally real fields, Math. Z. 134 (1973), 291–301. MR 330045, DOI 10.1007/BF01214693
- R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR 0230273
- Ju. L. Eršov, The number of linear orders on a field, Mat. Zametki 6 (1969), 201–211 (Russian). MR 249342
- John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
- Manfred Knebusch, Alex Rosenberg, and Roger Ware, Signatures on semilocal rings, J. Algebra 26 (1973), 208–250. MR 327761, DOI 10.1016/0021-8693(73)90021-5
- Manfred Knebusch, Alex Rosenberg, and Roger Ware, Structure of Witt rings and quotients of Abelian group rings, Amer. J. Math. 94 (1972), 119–155. MR 296103, DOI 10.2307/2373597
- Manfred Knebusch, Alex Rosenberg, and Roger Ware, Structure of Witt rings, quotients of abelian group rings, and orderings of fields, Bull. Amer. Math. Soc. 77 (1971), 205–210. MR 271091, DOI 10.1090/S0002-9904-1971-12683-6
- T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
- F. Lorenz and J. Leicht, Die Primideale des Wittschen Ringes, Invent. Math. 10 (1970), 82–88 (German). MR 266949, DOI 10.1007/BF01402972
- John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372, DOI 10.1007/978-3-642-88330-9
- George F. Simmons, Introduction to topology and modern analysis, McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London 1963. MR 0146625
- M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481. MR 1501905, DOI 10.1090/S0002-9947-1937-1501905-7
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 209 (1975), 225-235
- MSC: Primary 12D15; Secondary 10C05
- DOI: https://doi.org/10.1090/S0002-9947-1975-0379448-X
- MathSciNet review: 0379448