Inverse Galois Theory
Inverse Galois Theory is concerned with the question of which finite groups occur as Galois Groups over a given field. In particular, this includes the question of the structure and the representations of the absolute Galois group of K and also the questi
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Springer-Verlag Berlin Heidelberg GmbH
Gunter Malle • B. Heinrich Matzat
Inverse Galois Theory
Springer
Gunter Malle
FB Mathematikllnformatik Universitat Gesamthochschule Kassel Heinrich-Plett-Stra6e 40 D-34132 Kassel, Germany e-mail: [email protected] B. Heinrich Matzat
Interdisziplinares Zentrum rur Wissenschaftliches Rechnen Universitat Heidelberg 1m Neuenheimer Feld 368 D-69120 Heidelberg, Germany e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Malle. Gunter: Inverse galois theory / Gunter Malle ; B. Heinrich Matzat. (Springer Monographs in mathematics) ISBN 978-3-642-08311-2 ISBN 978-3-662-12123-8 (eBook) DOI 10.1007/978-3-662-12123-8
Mathematics Subject Classification (1991): 12-xx, 12F12, 20-xx
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e Springer-Verlag Berlin Heidelberg 1999
Originally published by Springer-Verlag Berlin Heidelberg New York in 1999 The use of general descriptive names, registered names, trademarks 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.
SPIN 11012450
44/3111-54321- Printed on acid-free paper
Preface
Inverse Galois Theory is concerned with the question which finite groups occur as Galois groups over a given field K. In particular this includes the question on the structure and the representations of the absolute Galois group of K and also the question about its finite epimorphic images, the so-called inverse problem of Galois theory. In all these areas important progress was made in the last few years, about which we want to report here. The first systematic approach to the solution of the inverse problem over the field of rational numbers Q goes back to Hilbert (1892). Using the irreducibility theorem which he proved for this purpose, he could show that over Q and more generally over every field finitely generated over Q there exist infinitely many Galois extensions with the symmetric and the alternating groups Sn and An. E. Noether (1918) then stated that the inverse problem for a finite group can be solved with the Hilbert irreducibility theorem if the field of fractions of the ring of invariants of a permutation representation of the group is rational, and that in this case all polynomials with thi
