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Structure of the Brauer ring of a field extension
- Source :
- Illinois J. Math. 52, no. 1 (2008), 261-277
- Publication Year :
- 2008
- Publisher :
- Duke University Press, 2008.
-
Abstract
- In 1986, Jacobson has defined the Brauer ring $B(E, D)$ for a finite Galois field extension $E/D$, whose unit group canonically contains the Brauer group of $D$. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group $G$ is trivial, the Brauer ring of the trivial extension $E/E$ becomes naturally isomorphic to the group ring of the Brauer group of $E$. In this paper, we generalize this result to any finite group $G$ via the theory of the restriction functor, by means of the well-understood functor $−_+$. More generally, we determine the structure of the $F$-Burnside ring for any additive functor $F$. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate $F$ related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension $ℂ/ℝ$.
- Subjects :
- Modular representation theory
Pure mathematics
Ring (mathematics)
Brauer's theorem on induced characters
Galois cohomology
General Mathematics
Mathematics::Rings and Algebras
Abelian extension
18A25
Algebra
Field extension
Mathematics::K-Theory and Homology
18A40
Mathematics::Representation Theory
Brauer group
Mathematics
Group ring
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Illinois J. Math. 52, no. 1 (2008), 261-277
- Accession number :
- edsair.doi.dedup.....b021656cb2d4f23408b550f1620d6a17