Diskette entnommen Aufstellung an Sonderstandort unter Buchsignatur Learning Abstract Algebra with ISETL Ed Dubinsky Uri Leron Learning Abstract Algebra with ISETL Macintosh™ Diskette Provided Springer-Verlag Berlin Heidelberg GmbH Ed Dubinsky Uri Leron Departments of Curriculum & Department of Science Education Instruction and Mathematics Technion Purdue University Israel Institute of Technology West Lafayette, IN 47907 32000 Haifa USA Israel With 2 Illustrations Mathematics Subject Classification (1991): 13-01, 20-01 Library of Congress Cataloging-in-Publication Data Dubinsky, Ed. Learning abstract algebra with ISETL I Ed Dubinsky, Uri Leron. p. cm. Includes bibliographical references and index. ISBN 978-3-540-94152-1 ISBN 978-3-662-25454-7 (eBook) DOI 10.1007/978-3-662-25454-7 I. Algebra, Abstract -Computer-assisted instruction. 2. ISETL (Computer program language) I. Leron, Uri. II. Title. QA162.D83 1993 512' .02'078-dc20 93-2609 Printed on acid-free paper. © 1994 Springer-Verlag Berlin Heidelberg Originally published by Springer-Verlag New York, Inc in 1994. Softcover reprint of the hardcover 1st edition 1994 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag Berlin Heidelberg GmbH), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf ter developed is forbidden. The programming language ISETL on the enclosed diskette is copyrighted by Gary Levin. It is being distributed herewith by permission of Gary Levin. The utility programs on the enclosed diskette are under copyright protection. Copying the enclosed diskette for the purpose of making a profit is forbidden. Before using the programs please consult the technical manuals provided by the manufacturer of the computer. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former 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 Natalie Johnson; manufacturing supervised by Vincent Scelta. Camera-ready copy prepared using the authors' LaTeX files. 987654321 Contents Comments for the Student xi Comments for the Instructor xvii Acknowledgments xxi 1 Mathematical Constructions in ISETL 1 1.1 Using ISETL ..... 1 1.1.1 Activities.............. 1 1.1.2 Getting started .......... . 5 1.1.3 Simple objects and operations on them 6 1.1.4 Control statements . . . . . . . . . . 7 1.1.5 Exercises............... 8 1.2 Compound objects and operations on them 11 1.2.1 Activities. 11 1.2.2 Tuples......... 14 1.2.3 Sets ......... . 15 1.2.4 Set and tuple formers 16 1.2.5 Set operations 17 1.2.6 Permutations..... 17 1.2.7 Quantification ... . 18 1.2.8 Miscellaneous ISETL features 20 1.2.9 VISETL 20 1.2.10 Exercises ........... . 21 vi Contents 1.3 Functions in ISETL 25 1.3.1 Activities.. 25 1.3.2 Funes.... 31 1.3.3 Alternative syntax for funcs .. 32 1.3.4 Using funcs to represent situations 33 1.3.5 Funes for binary operations . 33 1.3.6 Funes to test properties 33 1.3.7 Smaps.. 34 1.3.8 Procs .. 35 1. 3.9 Exercises 35 2 Groups 39 2.1 Getting acquainted with groups. 39 2.1.1 Activities...... 39 2.1.2 Definition of a group. 42 2.1.3 Examples of groups 43 Nu mber systems . 43 Integers mod n 45 Symmetric groups. 47 Symmetries of the square . 49 Groups of matrices .. .. 52 2.1.4 Elementary properties of groups 52 2.1.5 Exercises . . . . . . . . . . . . . 55 2.2 The modular groups and the symmetric groups 57 2.2.1 Activities......... 57 2.2.2 The modular groups Zn 60 2.2.3 The symmetric groups Sn 65 Orbits and cycles 68 2.2.4 Exercises . . 69 2.3 Properties of groups . . . . . 71 2.3.1 Activities... .. 71 2.3.2 The specific and the general. 72 2.3.3 The cancellation law-An illustration of the abstract method . . . . . . . . . . 74 2.3.4 How many groups are there? . . 75 Classifying groups of order 4 . 77 2.3.5 Looking ahead-subgroups .. . . 79 2.3.6 Summary of examples and non-examples of groups 80 2.3.7 Exercises . . . . . . . . . . . . . . . .. ...... 81 3 Subgroups 83 3.1 Definitions and examples. 83 3.1.1 Activities.... 83 3.1.2 Subsets of a group 86 Definition of a subgroup 86 Contents vii 3.1.3 Examples of subgroups. . . . . . . . . 88 Embedding one group in another 88 Conjugates . . . . . . . . . . . . . 89 Cycle decomposition and conjugates in Sn 91 3.1.4 Exercises.......... 92 3.2 Cyclic groups and their subgroups ......... . 94 3.2.1 Activities.................... 94 3.2.2 The subgroup generated by a single element . 96 3.2.3 Cyclic groups . . . . . . . 100 The idea of the proof 101 3.2.4 Generators .. . . . . . . 103 Generators of Sn .. 103 Parity-even and odd permutations. 104 Determining the parity of a permutation. . 105 3.2.5 Exercises... 105 3.3 Lagrange's theorem. . . . . . . . . . . . . . . 108 3.3.1 Activities................ 108 3.3.2 What Lagrange's theorem is all about 111 3.3.3 Cosets................. 112 3.3.4 The proof of Lagrange's theorem . . 113 3.3.5 Exercises............... 116 4 The FundaIllental Homomorphism Theorem 119 4.1 "Quotient groups ... . . 119 4.1.1 Activities................ 119 4.1.2 Normal subgroups . . . . . . . . . . . 121 Multiplying cosets by representatives 124 4.1.3 The quotient group . 125 4.1.4 Exercises 126 4.2 Homomorphisms . . . . . . 129 4.2.1 Activities...... 129 4.2.2 Homomorphisms and kernels 133 4.2.3 Examples........... 133 4.2.4 Invariants........... 135 4.2.5 Homomorphisms and normal subgroups 136 An interesting example 137 4.2.6 Isomorphisms. 138 4.2.7 Identifications..... 139 4.2.8 Exercises....... 141 4.3 The homomorphism theorem 143 4.3.1 Activities....... 143 4.3.2 The canonical homomorphism. . . 145 4.3.3 The fundamental homomorphism theorem. 147 4.3.4 Exercises................... 150 viii Contents 5 Rings 153 5.1 Rings .......... · . 153 5.1.1 Activities · ... 153 5.1.2 Definition of a ring . 156 5.1.3 Examples of rings 156 5.1.4 Rings with additional properties 157 Integral domains ...... 157 Fields .... · . · ...... 158 5.1.5 Constructing new rings from old-matrices 159 5.1.6 Constructing new rings from old-polynomials 161 5.1.7 Constructing new rings from old-functions 164 5.1.8 Elementary properties-arithmetic 165 5.1.9 Exercises 165 5.2 Ideals .......... · . · ....... 168 5.2.1 Activities · .. · . · ....... 168 5.2.2 Analogies between groups and rings 170 5.2.3 Subrings ..... · . · .. 171 Definition of subring ... 171 5.2.4 Examples of subrings ...... 171 Subrings of Zn and Z . . . 171 Subrings of M(R) ..... 172 Subrings of polynomial rings . 172 Subrings of rings of functions 173 5.2.5 Ideals and quotient rings. 173 Definition of ideal . . . . . 173 Examples of ideals .... 175 5.2.6 Elementary properties of ideals 175 5.2.7 Elementary properties of quotient rings 176 Quotient rings that are integral domains- prime ideals · . · .......... · ... 176 Quotient rings that are fields-maximal ideals . 177 5.2.8 Exercises · .. · . 178 • •••• I 5.3 Homomorphisms and isomorphisms . . . . . . . · .. 181 5.3.1 Activities · ... · . · .......... · .. 181 5.3.2 Definition of homomorphism and isomorphism 182 Group homomorphisms vs. ring homomorphisms 183 5.3.3 Examples of homomorphisms and isomorphisms. 183 Homomorphisms from Zn to Zk . 183 Homomorphisms of Z . . . . . . . . ... 184 Homomorphisms of polynomial rings · ... 184 Embeddings-Z, Zn as universal subobjects 184 The characteristic of an integral domain and a field ... · . · ..... 185 5.3.4 Properties of homorphisms .. 186 Preservation · . · ..... 186 Contents ix Ideals and kernels of ring homomorphisms 186 5.3.5 The fundamental homomorphism theorem 187 The canonical homomorphism . 187 The fundamental theorem ... 187 Homomorphic images of Z, Zn 188 Identification of quotient rings . 188 5.3.6 Exercises ... . . . 190 6 Factorization in Integral Domains 193 6.1 Divisibility properties of integers and polynomials. 193 6.1.1 Activities............ 193 6.1.2 The integral domains Z, Q[x] . . . . . . . 198 Arithmetic and factoring . . . . . . . 198 The meaning of unique factorization 199 6.1.3 Arithmetic of polynomials . . . . . 200 Long division of polynomials . . . . . 200 6.1.4 Division with remainder ......... 202 6.1.5 Greatest Common Divisors and the Euclidean algorithm . . . . . . . . . . . . 204 6.1.6 Exercises...... . . . . . . 208 6.2 Euclidean domains and unique factorization 209 6.2.1 Activities....... . . 209 6.2.2 Gaussian integers . . . . . . .. 212 6.2.3 Can unique factorization fail? . . . . 214 6.2.4 Elementary properties of integral domains . 214 6.2.5 Euclidean domains . . . . . . . . . . . 218 Examples of Euclidean domains . . . . 219 6.2.6 Unique factorization in Euclidean domains. 221 6.2.7 Exercises...... . . 225 6.3 The ring of polynomials over a field . 226 6.3.1 Unique factorization in F[x] . 227 6.3.2 Roots of polynomials . . . 228 6.3.3 The evaluation homomorphism . . . 230 6.3.4 Reducible and irreducible polynomials 231 Examples. . . . . ..... . 231 6.3.5 Extension fields. . . . ..... . 235 Construction of the complex numbers . 237 6.3.6 Splitting fields 237 6.3.7 Exercises...... . . . . . . . . . . 239 Index 241