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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Relative Brauer groups of discrete valued fields
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by Burton Fein and Murray Schacher PDF
Proc. Amer. Math. Soc. 127 (1999), 677-684 Request permission


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:
  • Murray Schacher
  • Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024
  • Email:
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 677-684
  • MSC (1991): Primary 12G05; Secondary 12E15
  • DOI:
  • MathSciNet review: 1487365