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Brauer Factor Sets, Noether Factor Sets, and Crossed Products

In: Emmy Noether in Bryn Mawr

Author

Listed:
  • Nathan Jacobson

    (Yale University, Department of Mathematics)

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.

Suggested Citation

  • Nathan Jacobson, 1983. "Brauer Factor Sets, Noether Factor Sets, and Crossed Products," Springer Books, in: Bhama Srinivasan & Judith D. Sally (ed.), Emmy Noether in Bryn Mawr, pages 1-20, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-5547-5_1
    DOI: 10.1007/978-1-4612-5547-5_1
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