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Relative Brauer Groups of Discrete Valued Fields

Burton Fein and Murray Schacher
Proceedings of the American Mathematical Society
Vol. 127, No. 3 (Mar., 1999), pp. 677-684
Stable URL: http://www.jstor.org/stable/118998
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Relative Brauer Groups of Discrete Valued Fields
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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, 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, Br(E/K)q, is non-trivial, (ii) Br(E/K)q is infinite, and (iii) there exists a valuation π of E trivial on k such that q divides the order of the decomposition group of E/K at π .

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