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Graduate Texts in Mathematics 204 Graduate Texts in Mathematics 204 Editorial Board S. Axler F.W. Gehring K.A. Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics TAKEUTIlZARING.lntroduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nded. 2 OxrOBY. Measure and Category. 2nd ed. 35 Al.ExANDERIWERMER. Several Complex 3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed. 2nded. 36 l<ELLEyINAMIOKA et al. Linear Topological 4 HILTON/STAMMBACH. A Course in Spaces. Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAc LANE. Categories for the Working 38 GRAUERT/FRlTZSCHE. Several Complex Mathematician. 2nd ed. Variables. 6 HUGHEs/Pn>ER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 SERRE. A Course in Arithmetic. 40 KEMENY/SNELLIKNAPP. Denumerable 8 T AKEUTIlZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and and Representation Theory. Dirichlet Series in Number Theory. 10 COHEN. A Course in Simple Homotopy 2nded. Theory. 42 SERRE. Linear Representations of Finite II CONWAY. Functions of One Complex Groups. Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSoN/FULLER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry. of Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. 14 GoLUBITSKy/GUILLEMIN. Stable Mappings 46 LoEVE. Probability Theory II. 4th ed. and Their Singularities. 47 MorSE. Geometric Topology in 15 BERBERIAN. Lectures in Functional Dimensions 2 and 3. Analysis and Operator Theory. 48 SACHs/WU. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGlWEIR. Linear Geometry. 18 HALMos. Measure Theory. 2nded. 19 HALMos. A Hilbert Space Problem Book. 50 EDWARDS. Fennat's Last Theorem. 2nded. 51 KLINGENBERG. A Course in Differential 20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry. 21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry. 22 BARNEs/MAcK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic. to Mathematical Logic. 54 GRAVERlWATKlNS. Combinatorics with 23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs. 24 HOLMES. Geometric Functional Analysis 55 BRoWN!PEARCY. Introduction to Operator and Its Applications. Theory I: Elements of Functional Analysis. 25 HEWITT/STROMBERG. Real and Abstract 56 MAsSEY. Algebraic Topology: An Analysis. Introduction. 26 MANEs. Algebraic Theories. 57 CRoWELLlFox. Introduction to Knot 27 l<ELLEy. General Topology. Theory. 28 ZARISKIISAMUEL. Commutative Algebra. 58 KOBLITZ. p-adic Numbers, p-adic Vol.I. Analysis, and Zeta-Functions. 2nd ed. 29 ZARIsKIlSAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields. Vol.Il. 60 ARNOLD. Mathematical Methods in 30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed. Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra II. Theory. Linear Algebra 62 KARGAPOLOv~AKov.Fundamenta1s 32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups. ill. Theory of Fields and Galois Theory. 63 BOLLOBAS. Graph Theory. 33 HIRsCH. Differential Topology. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. (continued after index) Jean-Pierre Escofier Galois Theory Translated by Leila Schneps With 48 Illustrations Springer Jean-Pierre Escofier Translator Institute Mathematiques de Rennes Leila Schneps Campus de Beaulieu 36 rue de ]' Orillon Universite de Rennes 1 75011 Paris 35042 Rennes Cedex France France [email protected] [email protected] Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hali University of California University University of Michigan at Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840 USA USA USA Mathematics Subject Classification (2000): 11 R32, Il S20, 12F10, 13B05 Library of Congress Cataloging-in-Publieation Data Eseofier, Jean-Pierre. Galois theory / Jean-Pierre Escofier. p. em. - (Graduate texts in mathematics; 204) Includes bibliographical references and index. ISBN 978-1-4612-6558-0 ISBN 978-1-4613-0191-2 (eBook) DOI 10.1007/978-1-4613-0191-2 1. Galois theory. 1. Title. II. Series. QA174.2 .E73 2000 5 12'.3-de2 1 00-041906 Printed on acid-free paper. Translated from the Freneh Theorie de Galois, by Jean-Pierre Eseofier, first edition published by Masson, Paris, © 1997, and second edition published by Dunod, Paris, © 2000, 5, rue Laromiguiere, 75005 Paris, France. © 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 Softcover reprint of the hardcover 1st edition 2001 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews ar scholarly analysis. Use in connection with any form of infor­ mation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the forrner are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Francine McNeill; manufacturing supervised by Joe Quatela. Photocomposed copy prepared from the translator' s TeX files. 9 8 7 654 3 2 1 ISBN 978-1-4612-6558-0 SPIN 1071 1904 Preface This book begins with a sketch, in Chapters 1 and 2, of the study of alge­ braic equations in ancient times (before the year 1600). After introducing symmetric polynomials in Chapter 3, we consider algebraic extensions of fi­ nite degree contained in the field <C of complex numbers (to remain within a familiar framework) and develop the Galois theory for these fields in Chapters 4 to 8. The fundamental theorem of Galois theory, that is, the Galois correspondence between groups and field extensions, is contained in Chapter 8. In order to give a rounded aspect to this basic introduction of Galois theory, we also provide • a digression on constructions with ruler and compass (Chapter 5), • beautiful applications (Chapters 9 and 10), and • a criterion for solvability of equations by radicals (Chapters 11 and 12). Many of the results presented here generalize easily to arbitrary fields (at least in characteristic 0), or they can be adapted to extensions of infinite degree. I could not write a book on Galois theory without some mention of the exceptional life of Evariste Galois (Chapter 13). The bibliography provides details on where to obtain further information about his life, as well as information on the moving story of Niels Abel. After these chapters, we introduce finite fields (Chapter 14) and separable extensions (Chapter 15). Chapter 16 presents two topics of current research: vi Preface firstly, the inverse Galois problem, which asks whether all finite groups occur as Galois groups of finite extensions of Q and which we treat explicitly in one very simple case, and secondly, a method for computing Galois groups that can be programmed on a computer. Most of the chapters contain exercises and problems. Some of the state­ ments are for practice, or are taken from past examinations; others suggest interesting results beyond the scope of the text. Some solutions are given completely, others are sketchy, and certain solutions that would involve mathematics beyond the scope of the text are omitted completely. Finally, this book contains a brief sketch of the history of Galois theory. I would like to thank the municipal library in Rennes for having allowed me to reproduce some fragments of its numerous treasures. The entire book was written with its student readers in mind, and with constant, careful consideration of the question of what these students will remember of it several years from now. lowe tremendous thanks to Annette Houdebine-Paugam, who helped me many times, and to Bernard Le Stum and Masson, who read the later versions of the text and suggested many corrections and alterations. Jean-Pierre Escofier May 1997 Contents Preface v 1 Historical Aspects of the Resolution of Algebraic Equations 1 1.1 Approximating the Roots of an Equation ..... 1 1.2 Construction of Solutions by Intersections of Curves 2 1.3 Relations with Trigonometry ....... 2 1.4 Problems of Notation and Terminology. 3 1.5 The Problem of Localization of the Roots 4 1.6 The Problem of the Existence of Roots. 5 1. 7 The Problem of Algebraic Solutions of Equations 6 Toward Chapter 2 ... 7 2 Resolution of Quadratic, Cubic, and Quartic Equations 9 2.1 Second-Degree Equations 9 2.1.1 The Babylonians 9 2.1.2 The Greeks . 11 2.1.3 The Arabs. 11 2.1.4 Use of Negative Numbers 12 2.2 Cubic Equations ........ 13 2.2.1 The Greeks ....... 13 2.2.2 Omar Khayyam and Sharaf ad Din at Tusi 13 2.2.3 Scipio del Ferro, Tartaglia, Cardan . 14 2.2.4 Algebraic Solution of the Cubic Equation . 15 2.2.5 First Computations with Complex Numbers. 16 2.2.6 Raffaele Bombelli ............... 17 viii Contents 2.2.7 Fran<;ois Viete . 18 2.3 Quartic Equations .. 18 Exercises for Chapter 2 19 Solutions to Some of the Exercises 22 3 Symmetric Polynomials 25 3.1 Symmetric Polynomials 25 3.1.1 Background... 25 3.1.2 Definitions ... 26 3.2 Elementary Symmetric Polynomials 27 3.2.1 Definition........... 27 3.2.2 The Product of the X - Xi; Relations Between Co- efficients and Roots .................. 27 3.3 Symmetric Polynomials and Elementary Symmetric Polyno- mials . 29 3.3.1 Theorem . 29 3.3.2 Proposition 31 3.3.3 Proposition 32 3.4 Newton's Formulas 32 3.5 Resultant of Two Polynomials .
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