Abstract algebra 2011 by rosenberger fine carstensen

De Gruyter Graduate Celine Carstensen Benjamin Fine Gerhard Rosenberger Abstract Algebra Applications to Galois Theory, Algebraic Geometry and Cryptography De Gruyter Mathematics Subject Classification 2010: Primary: 12-01, 13-01, 16-01, 20-01; Secondary: 01-01, 08-01, 11-01, 14-01, 94-01 This book is Volume 11 of the Sigma Series in Pure Mathematics, Heldermann Verlag ISBN 978-3-11-025008-4 e-ISBN 978-3-11-025009-1 Library of Congress Cataloging-in-Publication Data Carstensen, Celine Abstract algebra : applications to Galois theory, algebraic geometry, and cryptography / by Celine Carstensen, Benjamin Fine, and Gerhard Rosenberger p cm Ϫ (Sigma series in pure mathematics ; 11) Includes bibliographical references and index ISBN 978-3-11-025008-4 (alk paper) Algebra, Abstract Galois theory Geometry, Algebraic Crytography I Fine, Benjamin, 1948Ϫ II Rosenberger, Gerhard III Title QA162.C375 2011 5151.02Ϫdc22 2010038153 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de ” 2011 Walter de Gruyter GmbH & Co KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: AZ Druck und Datentechnik GmbH, Kempten ϱ Printed on acid-free paper Printed in Germany www.degruyter.com Preface Traditionally, mathematics has been separated into three main areas; algebra, analysis and geometry Of course there is a great deal of overlap between these areas For example, topology, which is geometric in nature, owes its origins and problems as much to analysis as to geometry Further the basic techniques in studying topology are predominantly algebraic In general, algebraic methods and symbolism pervade all of mathematics and it is essential for anyone learning any advanced mathematics to be familiar with the concepts and methods in abstract algebra This is an introductory text on abstract algebra It grew out of courses given to advanced undergraduates and beginning graduate students in the United States and to mathematics students and teachers in Germany We assume that the students are familiar with Calculus and with some linear algebra, primarily matrix algebra and the basic concepts of vector spaces, bases and dimensions All other necessary material is introduced and explained in the book We assume however that the students have some, but not a great deal, of mathematical sophistication Our experience is that the material in this can be completed in a full years course We presented the material sequentially so that polynomials and field extensions preceded an in depth look at group theory We feel that a student who goes through the material in these notes will attain a solid background in abstract algebra and be able to move on to more advanced topics The centerpiece of these notes is the development of Galois theory and its important applications, especially the insolvability of the quintic After introducing the basic algebraic structures, groups, rings and fields, we begin the theory of polynomials and polynomial equations over fields We then develop the main ideas of field extensions and adjoining elements to fields After this we present the necessary material from group theory needed to complete both the insolvability of the quintic and solvability by radicals in general Hence the middle part of the book, Chapters through 14 are concerned with group theory including permutation groups, solvable groups, abelian groups and group actions Chapter 14 is somewhat off to the side of the main theme of the book Here we give a brief introduction to free groups, group presentations and combinatorial group theory With the group theory material in hand we return to Galois theory and study general normal and separable extensions and the fundamental theorem of Galois theory Using this we present several major applications of the theory including solvability by radicals and the insolvability of the quintic, the fundamental theorem of algebra, the construction of regular n-gons and the famous impossibilities; squaring the circling, doubling the cube and trisecting an angle We vi Preface finish in a slightly different direction giving an introduction to algebraic and group based cryptography October 2010 Celine Carstensen Benjamin Fine Gerhard Rosenberger Contents Preface Groups, Rings and Fields 1.1 Abstract Algebra 1.2 Rings 1.3 Integral Domains and Fields 1.4 Subrings and Ideals 1.5 Factor Rings and Ring Homomorphisms 1.6 Fields of Fractions 1.7 Characteristic and Prime Rings 1.8 Groups 1.9 Exercises v 1 13 14 17 19 Maximal and Prime Ideals 2.1 Maximal and Prime Ideals 2.2 Prime Ideals and Integral Domains 2.3 Maximal Ideals and Fields 2.4 The Existence of Maximal Ideals 2.5 Principal Ideals and Principal Ideal Domains 2.6 Exercises 21 21 22 24 25 27 28 29 29 35 38 41 45 51 51 53 53 55 57 58 65 Prime Elements and Unique Factorization Domains 3.1 The Fundamental Theorem of Arithmetic 3.2 Prime Elements, Units and Irreducibles 3.3 Unique Factorization Domains 3.4 Principal Ideal Domains and Unique Factorization 3.5 Euclidean Domains 3.6 Overview of Integral Domains 3.7 Exercises Polynomials and Polynomial Rings 4.1 Polynomials and Polynomial Rings 4.2 Polynomial Rings over Fields 4.3 Polynomial Rings over Integral Domains 4.4 Polynomial Rings over Unique Factorization Domains 4.5 Exercises viii Contents 66 66 69 70 74 75 78 Field Extensions and Compass and Straightedge Constructions 6.1 Geometric Constructions 6.2 Constructible Numbers and Field Extensions 6.3 Four Classical Construction Problems 6.3.1 Squaring the Circle 6.3.2 The Doubling of the Cube 6.3.3 The Trisection of an Angle 6.3.4 Construction of a Regular n-Gon 6.4 Exercises 80 80 80 83 83 83 83 84 89 91 91 94 100 100 101 105 109 111 Splitting Fields and Normal Extensions 8.1 Splitting Fields 8.2 Normal Extensions 8.3 Exercises 113 113 115 118 Groups, Subgroups and Examples 9.1 Groups, Subgroups and Isomorphisms 9.2 Examples of Groups 9.3 Permutation Groups 9.4 Cosets and Lagrange’s Theorem 9.5 Generators and Cyclic Groups 9.6 Exercises 119 119 121 125 128 133 139 Field Extensions 5.1 Extension Fields and Finite Extensions 5.2 Finite and Algebraic Extensions 5.3 Minimal Polynomials and Simple Extensions 5.4 Algebraic Closures 5.5 Algebraic and Transcendental Numbers 5.6 Exercises Kronecker’s Theorem and Algebraic Closures 7.1 Kronecker’s Theorem 7.2 Algebraic Closures and Algebraically Closed Fields 7.3 The Fundamental Theorem of Algebra 7.3.1 Splitting Fields 7.3.2 Permutations and Symmetric Polynomials 7.4 The Fundamental Theorem of Algebra 7.5 The Fundamental Theorem of Symmetric Polynomials 7.6 Exercises ix Contents 10 Normal Subgroups, Factor Groups and Direct Products 10.1 Normal Subgroups and Factor Groups 10.2 The Group Isomorphism Theorems 10.3 Direct Products of Groups 10.4 Finite Abelian Groups 10.5 Some Properties of Finite Groups 10.6 Exercises 141 141 146 149 151 156 160 11 Symmetric and Alternating Groups 11.1 Symmetric Groups and Cycle Decomposition 11.2 Parity and the Alternating Groups 11.3 Conjugation in Sn 11.4 The Simplicity of An 11.5 Exercises 161 161 164 167 168 170 171 171 172 175 177 179 12 Solvable Groups 12.1 Solvability and Solvable Groups 12.2 Solvable Groups 12.3 The Derived Series 12.4 Composition Series and the Jordan–Hölder Theorem 12.5 Exercises 13 Groups Actions and the Sylow Theorems 13.1 Group Actions 13.2 Conjugacy Classes and the Class Equation 13.3 The Sylow Theorems 13.4 Some Applications of the Sylow Theorems 13.5 Exercises 180 180 181 183 187 191 14 Free Groups and Group Presentations 14.1 Group Presentations and Combinatorial Group Theory 14.2 Free Groups 14.3 Group Presentations 14.3.1 The Modular Group 14.4 Presentations of Subgroups 14.5 Geometric Interpretation 14.6 Presentations of Factor Groups 14.7 Group Presentations and Decision Problems 14.8 Group Amalgams: Free Products and Direct Products 14.9 Exercises 192 192 193 198 200 207 209 212 213 214 216 ... Cataloging-in-Publication Data Carstensen, Celine Abstract algebra : applications to Galois theory, algebraic geometry, and cryptography / by Celine Carstensen, Benjamin Fine, and Gerhard Rosenberger p cm... Chapter Groups, Rings and Fields 1.1 Abstract Algebra Abstract algebra or modern algebra can be best described as the theory of algebraic structures Briefly, an algebraic structure is a set S together... 978-3-11-025008-4 (alk paper) Algebra, Abstract Galois theory Geometry, Algebraic Crytography I Fine, Benjamin, 1948Ϫ II Rosenberger, Gerhard III Title QA162.C375 2011 5151.02Ϫdc22 2010038153