Elementary Abstract Algebra: Examples and Applications

785 Pages·2017·12.02 MB·English

Abstract Algebra:
Examples and Applications
J. Hill, C. Thron et al.
Elementary Abstract Algebra:
Examples and Applications
Supervising editors:
Justin Hill, Chris Thron
St. Philip’s College / Texas A&M University-Central Texas
Incorporating source materials by
Thomas Judson (Stephen F. Austin State University)
Dave Witte Morris and Joy Morris (University of Lethbridge)
A. J. Hildebrand (University of Illinois Urbana-Champaign)
With chapters by
Christy Douglass, Jennifer Lazarus, Mark Leech, Moses
Marmolejo, Adam McDonald, Katrina Smith, Johnny Watts,
David Weathers, Holly Webb (TAMU-CT)
and additional contributions by
Semi Harrison,Rachel McCoy, Khoi Tran (TAMU-CT)
November 5, 2021
This book is offered under the GNU Free Documentation License, Version
1.2, or any later version published by the Free Software Foundation; with
no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A
copy of the license is included in the appendix, entitled “GNU Free
Dcoumentation License”. The Set Theory and Functions chapters are
licensed under Creative Commons license
(Attribution-NonCommercial-ShareAlike 2.0).
2
Material from the 2012 version of ”Abstract Algebra, Theory and Appli-
cations” by Thomas Judson may be found throughout much of the book.
Attributions are found at the beginning of each chapter. Judson’s book is
covered by the GFDL license, and all chapters of the current book contain-
ing his work are covered by the same license. A current version of ”Abstract
Algebra, Theory and Applications” may be found at:
abstract.ups.edu.
The Set Theory and Functions chapters are largely based on material
from ”Proofs and Concepts”(version 0.78, May 2009) by Dave Witte Morris
and Joy Morris, which may be found online at:
https://archive.org/details/flooved3499, or
http://people.uleth.ca/~dave.morris/books/proofs+concepts.html
TheirbookiscoveredbytheCreativeCommonslicense(Attribution-NonCommercial-
ShareAlike 2.0), and the two chapters containing their work are covered by
the same license.
The material on induction was modified from LATEXcode originally ob-
tained from A. J. Hildebrand, whose course web page is at:
http://www.math.uiuc.edu/~hildebr/
Justin and Chris would like to express their deepest gratitude to Tom, Dave
and Joy, and A. J. for generously sharing their original material. They were
not involved in the preparation of this manuscript, and are not responsible
for any errors or other shortcomings.
Pleasesendcommentsandcorrectionsto: [e-mail]. Youmayalso
request the solutions manual and LATEX source code from this same email
address.
Course web page: http://abstractalgebra.altervista.org/
Online book: http://sl2x.aimath.org/book/aafmt/
©2013,2014, 2015, 2018, 2020 by Justin Hill and Chris Thron. Some
rights reserved.
Portions ©1997 by Thomas Judson. Some rights reserved.
3
Portions ©2006-2009 by Dave Witte Morris and Joy Morris. Some rights
reserved.
Permission is granted to copy, distribute and/or modify this document
underthetermsoftheGNUFreeDocumentationLicense,Version1.2orany
later version published by the Free Software Foundation; with no Invariant
Sections, no Front-Cover Texts, and no Back-Cover Texts. The license may
be found at: https://www.gnu.org/licenses/fdl-1.3.en.html
ISBN: 978-0-359-04211-1
Contents
1 Forward 1
2 Glossary of symbols 8
3 Preliminaries 10
3.1 In the Beginning . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Integers, rational numbers, real numbers . . . . . . . . . . . . 11
3.2.1 Properties of arithmetic operations . . . . . . . . . . . 12
3.2.2 Order relations . . . . . . . . . . . . . . . . . . . . . . 15
3.2.3 Manipulating equations and inequalities . . . . . . . . 16
3.2.4 Exponentiation (VERY important) . . . . . . . . . . . 20
3.3 Test yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Complex Numbers 25
4.1 The origin of complex numbers . . . . . . . . . . . . . . 25
4.1.1 A number that can’t be real (and we can prove it!) . . 25
4.1.2 Unreal, but unavoidable . . . . . . . . . . . . . . . . . 29
4.1.3 A mathematical revolution . . . . . . . . . . . . . . . 30
4.2 Arithmetic with complex numbers . . . . . . . . . . . . . 35
4.2.1 Complex arithmetic . . . . . . . . . . . . . . . . . . . 35
4
CONTENTS 5
4.2.2 Comparison of integer, rational, real and complex ad-
dition properties . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 Comparisonofinteger,rational,realandcomplexmul-
tiplication properties . . . . . . . . . . . . . . . . . . . 41
4.2.4 Modulus and complex conjugate . . . . . . . . . . . . 42
4.3 Alternative representations of complex numbers . . . . . 46
4.3.1 Cartesian representation of complex numbers . . . . . 46
4.3.2 Vector representation of complex numbers . . . . . . . 46
4.3.3 Polar representation of complex numbers . . . . . . . 48
4.3.4 Converting between rectangular and polar form . . . . 48
4.3.5 Multiplication and powers in complex polar form . . . 52
4.3.6 A Remark on representations of complex numbers . . 59
4.4 Complex numbers and roots of algebraic equations . . . . . . 60
4.4.1 Roots of unity and regular polygons . . . . . . . . 60
4.4.2 Complex nth roots in general . . . . . . . . . . . 67
4.4.3 Complex roots of polynomial equations . . . . . . 70
4.5 Applications of complex numbers . . . . . . . . . . . . . 74
4.5.1 General remarks on the usefulness of complex numbers 74
4.5.2 Complex numbers in electrical engineering: phasors . 74
4.5.3 Complex numbers and fractals: the Mandelbrot set . . 80
4.6 Hints for “Complex Numbers” exercises . . . . . . . . . . . . 84
4.7 Study guide for “Complex Numbers” chapter . . . . . . . . . 86
5 Modular Arithmetic 90
5.1 Introductory examples . . . . . . . . . . . . . . . . . . . 90
5.2 Modular equivalence and modular arithmetic . . . . . . . . . 92
5.3 Modular equations . . . . . . . . . . . . . . . . . . . . . 101
5.3.1 More uses of modular arithmetic . . . . . . . . . . . . 101
5.3.2 Solving modular equations. . . . . . . . . . . . . . . . 104
5.4 The integers mod n (a.k.a. Z ) . . . . . . . . . . . . . . 111
n
5.4.1 Remainder arithmetic . . . . . . . . . . . . . . . . . . 111
6 CONTENTS
5.4.2 Cayley tables for Z . . . . . . . . . . . . . . . . . . . 117
n
5.4.3 Closure properties of Z . . . . . . . . . . . . . . . . . 119
n
5.4.4 Identities and inverses in Z . . . . . . . . . . . . . . . 121
n
5.4.5 Inverses in Z . . . . . . . . . . . . . . . . . . . . . . . 121
n
5.4.6 Other arithmetic properties of ⊕ and (cid:12) . . . . . . . . 123
5.4.7 Group: a central concept in abstract algebra . . . . . 124
5.5 Modular division . . . . . . . . . . . . . . . . . . . . . . 126
5.5.1 A sticky problem . . . . . . . . . . . . . . . . . . . . . 126
5.5.2 Greatest common divisors . . . . . . . . . . . . . . . . 131
5.5.3 Computer stuff . . . . . . . . . . . . . . . . . . . . . . 135
5.5.4 Diophantine equations . . . . . . . . . . . . . . . . . . 136
5.5.5 Multiplicative inverse for modular arithmetic . . . . . 144
5.5.6 Chinese remainder theorem . . . . . . . . . . . . . . . 147
5.6 Hints for “Modular Arithmetic” exercises . . . . . . . . . . . 150
5.7 Study guide for “Modular Arithmetic” chapter . . . . . . . . 152
6 Modular Arithmetic, Decimals, and Divisibility 156
6.1 Decimal representations . . . . . . . . . . . . . . . . . . . . . 156
6.1.1 Decimal representation formula . . . . . . . . . . . . . 156
6.1.2 Formulas for decimal digits of of integers . . . . . . . 157
6.1.3 Formulas for decimal digits of nonintegers . . . . . . . 159
6.1.4 Repeating decimals . . . . . . . . . . . . . . . . . . . . 162
6.1.5 Divisibility rules . . . . . . . . . . . . . . . . . . . . . 164
6.2 Decimal representations in other bases . . . . . . . . . . . . . 168
7 Set Theory 174
7.1 Set Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.1.1 Definition and examples . . . . . . . . . . . . . . . . . 175
7.1.2 Important sets of numbers . . . . . . . . . . . . . . . . 178
7.1.3 Operations on sets . . . . . . . . . . . . . . . . . . . . 180
7.2 Properties of set operations . . . . . . . . . . . . . . . . 186
CONTENTS 7
7.3 Do the subsets of a set form a group? . . . . . . . . . . 192
7.4 Hints for “Set Theory” exercises . . . . . . . . . . . . . . . . 195
7.5 Study guide for “Set Theory” chapter . . . . . . . . . . . . . 196
8 Functions: Basic Concepts 198
8.1 The Cartesian product: a different type of set operation 198
8.2 Introduction to functions . . . . . . . . . . . . . . . . . 201
8.2.1 Informal look at functions . . . . . . . . . . . . . . . . 201
8.2.2 Official definition of functions . . . . . . . . . . . . . . 208
8.3 One-to-one functions . . . . . . . . . . . . . . . . . . . . 212
8.3.1 Concept and definition . . . . . . . . . . . . . . . . . . 212
8.3.2 Proving that a function is one-to-one . . . . . . . . . . 215
8.4 Onto functions . . . . . . . . . . . . . . . . . . . . . . . 224
8.4.1 Concept and definition . . . . . . . . . . . . . . . . . . 224
8.4.2 Proving that a function is onto . . . . . . . . . . . . . 226
8.5 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.5.1 Concept and definition . . . . . . . . . . . . . . . . . . 232
8.5.2 Proving that a function is a bijection . . . . . . . . . . 233
8.6 Composition of functions . . . . . . . . . . . . . . . . . . 239
8.6.1 Concept and definition . . . . . . . . . . . . . . . . . . 239
8.6.2 Proofs involving function composition . . . . . . . . . 244
8.7 Inverse functions . . . . . . . . . . . . . . . . . . . . . . 249
8.7.1 Concept and definition . . . . . . . . . . . . . . . . . . 249
8.7.2 Which functions have inverses? . . . . . . . . . . . . . 252
8.8 Do functions from A to B form a group? . . . . . . . . . . . . 255
8.9 Hints for “Functions: basic concepts” exercises . . . . . . . . 258
8.10 Study guide for “Functions: Basic Concepts” chapter . . . . . 259
8 CONTENTS
9 Introduction to Cryptography 264
9.1 Overview and basic terminology. . . . . . . . . . . . . . . . . 264
9.2 Private key cryptography . . . . . . . . . . . . . . . . . . . . 266
9.2.1 Shift codes . . . . . . . . . . . . . . . . . . . . . . . . 266
9.2.2 Affine codes . . . . . . . . . . . . . . . . . . . . . . . . 269
9.2.3 Monoalphabetic codes . . . . . . . . . . . . . . . . . . 272
9.2.4 Polyalphabetic codes . . . . . . . . . . . . . . . . . . . 273
9.2.5 Spreadsheet exercises. . . . . . . . . . . . . . . . . . . 276
9.3 Public key cryptography . . . . . . . . . . . . . . . . . . . . . 280
9.3.1 The RSA cryptosystem . . . . . . . . . . . . . . . 281
9.3.2 Message verification . . . . . . . . . . . . . . . . . . . 284
9.3.3 RSA exercises . . . . . . . . . . . . . . . . . . . . 285
9.3.4 Additional exercises: identifying prime numbers . 288
9.4 References and suggested readings . . . . . . . . . . . . . . . 295
9.5 Hints for “Applications (I): Introduction to Cryptography”
exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
9.6 Study guide for “Applications (I): Introduction to Cryptog-
raphy” chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10 Sigma Notation 299
10.1 Lots of examples . . . . . . . . . . . . . . . . . . . . . . 299
10.2 Algebraic rules for Sigmas . . . . . . . . . . . . . . . . . . . . 302
10.2.1 Constant multiples, sums, and products of sums . . . 302
10.3 Change of variable and rearrangement of sums . . . . . . 304
10.4 Common Sums . . . . . . . . . . . . . . . . . . . . . . . 312
10.5 Summation by parts . . . . . . . . . . . . . . . . . . . . . . . 316
11 Application: Sigma Notation in Linear Algebra 322
11.1 Introduction to sigma notation in linear algebra . . . . . 322
11.2 Matrix multiplication. . . . . . . . . . . . . . . . . . . . . . . 323
11.3 The identity matrix and the Kronecker delta . . . . . . . . . 326
CONTENTS 9
11.4 Abbreviated matrix notations . . . . . . . . . . . . . . . . . . 331
11.5 Matrix transpose and matrix inverse . . . . . . . . . . . . . . 333
11.5.1 Matrix transpose . . . . . . . . . . . . . . . . . . . . . 333
11.5.2 Matrix inverse . . . . . . . . . . . . . . . . . . . . . . 335
11.6 Rotation matrices (in 3 dimensions) . . . . . . . . . . . . . . 335
11.7 Matrix traces . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
11.8 Levi-Civita symbols and applications . . . . . . . . . . . . . . 341
11.8.1 Levi-Civita symbols: definitions and examples. . . . . 341
11.8.2 Levi-Civita symbols and determinants . . . . . . . . . 344
11.8.3 Levi-Civita symbols and cross products . . . . . . . . 349
11.8.4 Proof of the vector BAC-CAB Rule . . . . . . . . . . 352
11.8.5 Proof of Euler’s Rotation Theorem . . . . . . . . . . . 356
11.9 Hints for “Sigma Notation” and “Applications of Sigma No-
tation” exercises . . . . . . . . . . . . . . . . . . . . . . . . . 362
11.10Study guide for “Sigma Notation” chapter . . . . . . . . . . . 364
12 Polynomials 368
12.1 Why study polynomials? . . . . . . . . . . . . . . . . . . . . . 368
12.2 Review of polynomial arithmetic . . . . . . . . . . . . . . 371
12.3 Polynomial operations in summation notation . . . . . . . . . 373
12.4 More exotic polynomials . . . . . . . . . . . . . . . . . . . . . 380
12.5 Polynomial properties and summation notation . . . . . . . . 386
12.6 Polynomials and division . . . . . . . . . . . . . . . . . . . . . 394
12.6.1 The Division Algorithm for polynomials over fields 394
12.6.2 Greatest common divisors of polynomials . . . . . . . 398
12.6.3 Polynomial roots and the FTOA (easy part) . . 401
12.6.4 Algebraic closure and the FTOA (hard part) . . . . . 408
12.7 Hints for “Polynomial Rings” exercises . . . . . . . . . . . . . 413