Illinois Journal of Mathematics

Structure of the Brauer ring of a field extension

Hiroyuki Nakaoka

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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 $ℂ/ℝ$.

Article information

Illinois J. Math., Volume 52, Number 1 (2008), 261-277.

First available in Project Euclid: 15 May 2009

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Zentralblatt MATH identifier

Primary: 18A25: Functor categories, comma categories 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)


Nakaoka, Hiroyuki. Structure of the Brauer ring of a field extension. Illinois J. Math. 52 (2008), no. 1, 261--277. doi:10.1215/ijm/1242414131.

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