General Galois Geometries
This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second vol
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J.W.P. Hirschfeld J.A. Thas
General Galois Geometries
Springer Monographs in Mathematics
More information about this series at http://www.springer.com/series/3733
J.W.P. Hirschfeld • J.A. Thas
General Galois Geometries
J.W.P. Hirschfeld Department of Mathematics University of Sussex Brighton, UK
J.A. Thas Department of Mathematics Ghent University Gent, Belgium
ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-1-4471-6788-4 ISBN 978-1-4471-6790-7 (eBook) DOI 10.1007/978-1-4471-6790-7 Library of Congress Control Number: 2016930023 Mathematics Subject Classification (2010): 51E20, 51E12, 51E14, 51E22, 51E23, 94B05, 05B25, 14H50, 14L35, 14M15, 51A05, 51A30, 51A45, 51A50, 51D20, 51E21 Springer London Heidelberg New York Dordrecht © Springer-Verlag London 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer-Verlag London Ltd. is part of Springer Science+Business Media (www.springer.com)
For Lucca and Indigo JWPH For Sam, Jo and Maya JAT
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
1
Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Canonical forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Tangency and polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Numbers of subspaces on a quadric . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The orthogonal groups . . . . . . . . . . . . . . . . . . . . . . . . .
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