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

 

Relative Brauer groups
of discrete valued fields


Authors: Burton Fein and Murray Schacher
Journal: Proc. Amer. Math. Soc. 127 (1999), 677-684
MSC (1991): Primary 12G05; Secondary 12E15
MathSciNet review: 1487365
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Abstract: Let $E$ be a non-trivial finite Galois extension of a field $K$. In this paper we investigate the role that valuation-theoretic properties of $E/K$ play in determining the non-triviality of the relative Brauer group, $\operatorname{Br}(E/K)$, of $E$ over $K$. In particular, we show that when $K$ is finitely generated of transcendence degree 1 over a $p$-adic field $k$ and $q$ is a prime dividing $[E:K]$, then the following conditions are equivalent: (i) the $q$-primary component, $\operatorname{Br}(E/K)_{q}$, is non-trivial, (ii) $\operatorname{Br}(E/K)_{q}$ is infinite, and (iii) there exists a valuation $\pi $ of $E$ trivial on $k$ such that $q$ divides the order of the decomposition group of $E/K$ at $\pi $.


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Additional Information

Burton Fein
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: fein@math.orst.edu

Murray Schacher
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024
Email: mms@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04792-9
PII: S 0002-9939(99)04792-9
Keywords: Brauer group, discrete valued field
Received by editor(s): June 23, 1997
Additional Notes: The authors are grateful for support under NSA Grants MDA904-95-H-1054 and MDA904-95-H-1022, respectively.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1999 American Mathematical Society