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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

Murray Schacher
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024

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

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