The Schur Subgroup of the Brauer Group
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397 Toshihiko Yamada Tokyo Metropolitan University, Fukazawa-Cho Setagaya, Tokyo/Japan
The Schur Subgroup of the Brauer Group
Springer-Verl ag Berlin· Heidelberg· NewYork 1974
AMS Subject Classifications (1970): Primary: 20C05, 20C15 Secondary: 12A99, 12B99 ISBN 3-540-06806-6 Springer-Verlag Berlin· Heidelberg ·New York ISBN 0-387-06806-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under§ 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg "1974. Library of Congress Catalog Card Number 74-9104. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE These notes are taken from a course on theory of the Schur subgroup of the Brauer group which I gave at Queen's University in 1971/72, including subsequent developments of it. The study of the Schur subgroup or Schur algebra was begun by I. Schur at the beginning of this century. But it was not until 1945 that the long surmised conjecture: "Every irreducible representation
U
of a finite group
G
of order
n
can be written
in the field of' the n-th roots of unity," was solved by R.Brauer. Early in 1950's, R. Brauer and E. Witt, independently, found that questions on the Schur subgroup are reduced to a treatment f'or a cyclotomic algebra.
The result has been called the Brauer-
Witt theorem, and we can now say that almost all detailed results about Schur subgroups depend on it. After the Brauer-Witt theorem appeared, there was little progress until the end of' 1960's.
However, in a couple of years,
the Schur subgroup has been extensively studied by many people, and it seems that the development has reached a certain culminating point with the most recent results. some of them:
Here we will mention
The Schur subgroup was completely determined f'or
an arbitrary local field;
a simple formula f'or the index of a
p-adic cyclotomic algebra was obtained;
the Schur subgroups
are determined for several cyclotomic extensions of' the rational field
Q;
discovered;
some remarkable properties of' a Schur algebra were etc.
Thus it seems timely to undertake the task
IV of clarifying the theory of Schur subgroup systematically, focussing on the recent progress. I would like to thank Professor P. Ribenboim who let me give a course on this subject at Queen's University and suggested editing these notes for publication. I am also grateful to McGill University, Queen's University, Oregon State University, and the National Science Foundation, for financial assistance at various stages of this work.
TABLE OF CONTENTS
Introduction
l
Chapter l .
Schur algebras
4
Chapter 2.
Cyclotomic algebra
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