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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The linear and quadratic Schur subgroups over the $ S$-integers of a number field


Author: Carl R. Riehm
Journal: Proc. Amer. Math. Soc. 107 (1989), 83-87
MSC: Primary 11R65; Secondary 11R54, 13A20, 20C10
MathSciNet review: 979218
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Abstract: Let $ K$ be an algebraic number field and let $ \mathfrak{O}$ be a ring of $ S$-integers in $ K$ (where $ S$ is a set of primes of $ K$ containing all the archimedean primes); that is to say, $ \mathfrak{O}$ is a Dedekind domain whose field of quotients is $ K$. In analogy with a theorem of T. Yamada in the case of a field of characteristic 0, it is shown that if $ S\left( \mathfrak{O} \right)$ is the Schur subgroup of the Brauer group $ B\left( \mathfrak{O} \right)$ and if $ \mathfrak{o} = \mathfrak{O} \cap k$, where $ k$ is any field containing the maximal abelian extension of $ \mathbb{Q}$ in $ K$, then $ S\left( \mathfrak{O} \right) = \mathfrak{O} \otimes S\left( \mathfrak{o} \right)$, i.e. the Brauer classes in $ S\left( \mathfrak{O} \right)$ are those obtained from $ S\left( \mathfrak{o} \right)$ by extension of the scalars to $ \mathfrak{O}$. A similar theorem is proved as well in the case of the Schur subgroup $ S\left( {\mathfrak{O},\omega } \right)$ of the quadratic Brauer group $ B\left( {\mathfrak{O},\omega } \right)$, where $ \omega $ is an involution of $ \mathfrak{O}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0979218-0
PII: S 0002-9939(1989)0979218-0
Keywords: Schur subgroup, integral Brauer group, integral representations, representations of finite groups, quadratic Brauer group, Azumaya algebras, algebraic number fields
Article copyright: © Copyright 1989 American Mathematical Society