Brauer Factor Sets, Noether Factor Sets, and Crossed Products

  • Nathan Jacobson
Conference paper

Abstract

The role of Noether’s crossed products and factor sets in the study of the Brauer group Br(F) of a field F is well known. In particular, it is central in the determination of the Brauer group of a number field and in the proof of the Albert—Brauer—Hasse—Noether theorem that central division algebras over number fields are cyclic ([2], [5], [8], [9]). The central algebraic result of Noether’s theory is the isomorphism of the subgroup Br(E/F) of Br(F) consisting of the algebra classes having a finite dimensional Galois extension field E/F as a splitting field with the co-homology group H 2 (G, E*) where G = Gal E/F. This leads to an isomorphism (given later) of the full Brauer group Br(F) with a cohomology group of the Galois group of the separable algebraic closure of F.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. A. Albert. The structure of matrices with any normal division algebra of multiplications. Ann. Math., 32 (1931), 131–148.CrossRefGoogle Scholar
  2. [2]
    A. A. Albert and H. Hasse. A determination of all normal division algebras over an algebraic number field. Trans. Amer. Math. Soc., 34 (1932), 722–726.CrossRefMathSciNetGoogle Scholar
  3. [3]
    R. Brauer. Untersuchungen über die arithmetischen Eigenschaften von Gruppen linearer Substitutionen. Math. Z., 8 (1928), 677–696.CrossRefMathSciNetGoogle Scholar
  4. [4]
    —. On normal division algebras of degree 5. Proc. Nat. Acad. Sei., 24 (1938), 243–246.CrossRefGoogle Scholar
  5. [5]
    R. Brauer, H. Hasse and E. Noether. Beweis eines Hauptaatzes in der Theorie der Algebren. J. Math., 167 (1931), 399–404.Google Scholar
  6. [6]
    R. Brauer and E. Noether. Uber minimale Zerfallungskorper irreduzibler Darstellungen. Sitzungsher. Preuss. Akad. Wiss., 32 (1927), 221–228.Google Scholar
  7. [7]
    C. W. Curtis and I. Reiner. Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers: New York, London.Google Scholar
  8. [8]
    H. Hasse. Theory of cyclic algebras over an algebraic number field, Trans. Amer. Math. Soc., 34 (1932), 171–214. Also Additional additional note to the authors: “Theory of cyclic algebras over an algebraic number field.” ibid, 727–730.CrossRefMathSciNetGoogle Scholar
  9. [9]
    —. Die Struktur der R. Brauerschen Algebren-Klassengresppen. Math. Ann., 107 (1933), 731–760.CrossRefMathSciNetGoogle Scholar
  10. [10]
    N. Jacobson. The Theory of Rings. American Mathematical Society Surveys, 1943.MATHGoogle Scholar
  11. [11]
    N. Jacobson. Generation of separable and central simple algebras.J. Math., 36 (1957), 217–227.MATHGoogle Scholar
  12. [12]
    — Structure and Representations of Jordan Algebras. Ann. Math. Soc. Colloquium Publ., 1968.MATHGoogle Scholar
  13. [13]
    N. Jacobson. Basic Algebra II. W. H. Freeman and Company: San Francisco, 1980.MATHGoogle Scholar
  14. [14]
    —. Gesamelte Abhandlungen. Springer-Verlag: Berlin, Heidelberg, New York, 1982.Google Scholar
  15. [15]
    H. Weyl. Generalized Riemann matrices and factor sets. Ann. Math., 37 (1936), 709–745.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  • Nathan Jacobson
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations