Learning Abstract Algebra with ISETL

264 Pages·1994·11.97 MB·English

by  Ed Dubinsky

Learning Abstract Algebra with ISETL
Springer
New York
Berlill
Heidelberg
Bareelolla
Budapest
Hong Kong
London
Mila
II
Paris
Santa Clara
Singapore
Tokyo
Ed Dubinsky Uri Leron
Learning Abstract
Algebra with ISETL
DOS Diskette Provided
, Springer
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 / Ed Dubinsky, Uri Leron.
p. cm.
Includes bibliographical references and index.
Learning abstract algebra with ISETL / Ed Dubinsky, Uri Leron.
Additional Material to this bookcan be downloaded from http://extras.springer.com
ISBN-13:978-1-4612-7602-9 e-ISBN-13:978-1-4612-2602-4
D01:10.1007/978-1-4612-2602-4
I. Algebra, Abstract - Computer-assisted instruction. 2. ISETL
(Computer program language) I. Leron, Uri. II. Title.
QA162.D83 1993
S12'.02'078-dc20 93-2609
Printed on acid-free paper.
© 1994 Springer-Verlag New York, Inc.
Reprint of the original 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 New York, Inc., 175 Fifth Avenue, New
York, NY 10010, USA), 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 jistributed 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 manufacture.r of the computer.
The use of general descriptive m:.mes, 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.
9 8 7 6 5 4 3 2 (Corrected second printing. 1998)
ISBN-I3:978-1-4612-7602-9 SPIN 10662896
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 Funcs .. . . 31
1.3.3 Alternative syntax for funcs . 32
1.3.4 Using funcs to represent situations 33
1.3.5 Funcs for binary operations 33
1.3.6 Funcs 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
Number 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 Fundamental 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
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 . . . . . . . . . . . . 18. 4
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