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The Schur Subgroup of a p-Adic Field

Published online by Cambridge University Press:  20 November 2018

Eugene Spiegel  and
Allan Trojan
Eugene Spiegel
Affiliation:
University of Connecticut, Storrs, Connecticut
Allan Trojan
Affiliation:
Atkinson College, York University, Downsview, Ontario
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Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only if

where c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 300 - 303
Copyright
Copyright © Canadian Mathematical Society 1979

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