Skip to main content
SpringerLink
Log in

Lokal-Global-Prinzipien für die Brauergruppe

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

This article looks at Local-Global-Principles for the Brauer group, modeled after the celebrated theorem of Hasse-Brauer-Noether for the Brauer group of a number field. Pop introduced a property for fields, which holds especially for real closed andp-adically closed fields and yields a Local-Global-Principle for function fields of one variable over such fields. Then he used model theoretical means to generalize these results to arbitrary extensions of transcendental degree one over real closed andp-adically closed fields. This paper achieves this in a more elementary manner. Another result are examples of fields where the Local-Global-Principle is violated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. N. Bourbaki, Algèbre, Chapitre VIII, Paris, 1958

  2. J.W.S. Cassels, A. Fröhlich, Algebraic number theory, Washington D.C., 1967

  3. O. Endler, Valuation Theory, Berlin 1972

  4. M. Fried, M. Jarden, Field Arithmetic, Berlin 1986

  5. Hartshorne, Algebraic Geometry. Springer, Berlin Heidelberg, 1977

    MATH  Google Scholar 

  6. H. Hasse, R. Brauer und E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren. Journal für Mathematik, Band 167, 399–404

  7. U. Jensen, H. Lenzing, Model theoretic algebra, Amsterdam, 1989

  8. S. Lang, Abelian Varieties, New York, Interscience 1959

    Google Scholar 

  9. S. Lang, On quasialgebraic closure, Annals of Mathematics55 (1952), 373–390

    Article  MathSciNet  Google Scholar 

  10. S. Lichtenbaum, Duality Theorems for curves overp-adic fields, Invent. math.7 (1969), 120–136

    Article  MathSciNet  Google Scholar 

  11. O.T. O'Meara, Introduction to quadratic forms. Springer, Berlin Göttingen Heidelberg, 1963

    MATH  Google Scholar 

  12. F. Pop, Galoissche Kennzeichnungp-adisch abgeschlossener Körper, Journal reine angewandte Mathematik392 (1988), 145–175

    MathSciNet  Google Scholar 

  13. A. Prestel, P. Roquette, Formallyp-adic fields, Lecture Notes in Mathematics 1050, Berlin-Heidelberg-New York, 1984

  14. B.v. Querenburg, Mengentheoretische Topologie, Berlin, Heidelberg, New York, 1979

  15. P. Roquette, On the Galois Cohomology of the Projective Linear Group and its Applications to the Construction of Generic Splitting Fields of Algebras, Math. Annalen150 (1963), 411–439

    Article  MathSciNet  Google Scholar 

  16. J.P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics5, Berlin-Heidelberg-New York 1973

  17. J.P. Serre, Local Fields, New York, 1979

  18. C. Tsen, Divisionsalgebren über Funktionenkörper, Nachr. Ges. Wiss. Göttingen (1933), 335

  19. E. Witt, Zerlegung reeller algebraischer Funktionen in Quadrate, Schiefkörper über reellen Funktionenkörpern, Journal reine angewandte Mathematik171 (1934), 4–11

    Google Scholar 

  20. E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Zeitschrift39 (1935), 462–467

    Article  MathSciNet  Google Scholar 

  21. O. Zariski, P. Samuel, Commutative Algebra, New Jersey, 1960

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054, Erlangen

    Götz Wiesend

Authors
  1. Götz Wiesend
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This article was processed by the author using the IATEX style filecljour1 from Springer-Verlag.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wiesend, G. Lokal-Global-Prinzipien für die Brauergruppe. Manuscripta Math 86, 455–466 (1995). https://doi.org/10.1007/BF02568005

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02568005

Search

Navigation