Elementary Abstract Algebra- Examples and Applications, 2019a

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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) October 8, 2020 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 Applications” 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 containing 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 Their book is covered by the Creative Commons license (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 L A TEXcode originally obtained 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. Please send comments and corrections to: thron@tamuct.edu. You may also request the solutions manual and L A TEX sourcecodefromthissameemail address. Course web page: http://abstractalgebra.altervista.org/ Online book: http://sl2x.aimath.org/book/aafmt/ c○2013,2014, 2015, 2018, 2020 by Justin Hill and Chris Thron. Some rights reserved. Portions c○1997 by Thomas Judson. Some rights reserved.
Page 1: Abstract Algebra: Examples and Appl
Page 5 and 6: Contents Forward 1 To the student:
Page 7 and 8: 6 CONTENTS 3 Modular Arithmetic 90
Page 9 and 10: 8 CONTENTS 6.7.1 Concept and defini
Page 11 and 12: 10 CONTENTS 10 Symmetries of Plane
Page 13 and 14: 12 CONTENTS 14 Equivalence Relation
Page 15 and 16: 14 CONTENTS 18 Homomorphisms of Gro
Page 17 and 18: 16 CONTENTS 21.8 Non-formula induct
Page 19 and 20: 2 CONTENTS But students who don’t
Page 21 and 22: 4 CONTENTS • The book’s web sit
Page 23 and 24: Organization Plan of the Book A cha
Page 25 and 26: 8 CONTENTS connect algebraic aspect
Page 27 and 28: Glossary of Symbols cis θ: cosθ +
Page 29 and 30: 1 Preliminaries 1.1 In the Beginnin
Page 31 and 32: 14 CHAPTER 1 PRELIMINARIES multipli
Page 33 and 34: 16 CHAPTER 1 PRELIMINARIES (a) (((x
Page 35 and 36: 18 CHAPTER 1 PRELIMINARIES (b) a
Page 37 and 38: 20 CHAPTER 1 PRELIMINARIES think ba
Page 39 and 40: 22 CHAPTER 1 PRELIMINARIES Reason P
Page 41 and 42: 24 CHAPTER 1 PRELIMINARIES Exercise
Page 43 and 44: 2 Complex Numbers horatio: O day an
Page 45 and 46: 28 CHAPTER 2 COMPLEX NUMBERS • In
Page 47 and 48: 30 CHAPTER 2 COMPLEX NUMBERS 2.1.2
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36 CHAPTER 2 COMPLEX NUMBERS What c
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38 CHAPTER 2 COMPLEX NUMBERS Exerci
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40 CHAPTER 2 COMPLEX NUMBERS (a) (3
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42 CHAPTER 2 COMPLEX NUMBERS Additi
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44 CHAPTER 2 COMPLEX NUMBERS so tha
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46 CHAPTER 2 COMPLEX NUMBERS (a) Sh
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48 CHAPTER 2 COMPLEX NUMBERS 2.3.2
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50 CHAPTER 2 COMPLEX NUMBERS We kno
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52 CHAPTER 2 COMPLEX NUMBERS (a) Fi
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54 CHAPTER 2 COMPLEX NUMBERS ♦ We
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56 CHAPTER 2 COMPLEX NUMBERS (a) 2
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58 CHAPTER 2 COMPLEX NUMBERS If (r
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60 CHAPTER 2 COMPLEX NUMBERS 2.3.6
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62 CHAPTER 2 COMPLEX NUMBERS To ill
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64 CHAPTER 2 COMPLEX NUMBERS ♦ Ex
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66 CHAPTER 2 COMPLEX NUMBERS B · A
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68 CHAPTER 2 COMPLEX NUMBERS (a) Gi
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70 CHAPTER 2 COMPLEX NUMBERS One so
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72 CHAPTER 2 COMPLEX NUMBERS The
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74 CHAPTER 2 COMPLEX NUMBERS (a) Gi
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76 CHAPTER 2 COMPLEX NUMBERS Figure
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78 CHAPTER 2 COMPLEX NUMBERS Exerci
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80 CHAPTER 2 COMPLEX NUMBERS (a) Us
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82 CHAPTER 2 COMPLEX NUMBERS The Ma
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84 CHAPTER 2 COMPLEX NUMBERS 2.6 Hi
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86 CHAPTER 2 COMPLEX NUMBERS 2.7 St
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88 CHAPTER 2 COMPLEX NUMBERS Sectio
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3 Modular Arithmetic What goes up,
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92 CHAPTER 3 MODULAR ARITHMETIC Not
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94 CHAPTER 3 MODULAR ARITHMETIC On
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96 CHAPTER 3 MODULAR ARITHMETIC □
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98 CHAPTER 3 MODULAR ARITHMETIC Sup
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100 CHAPTER 3 MODULAR ARITHMETIC wa
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102 CHAPTER 3 MODULAR ARITHMETIC 0
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104 CHAPTER 3 MODULAR ARITHMETIC (i
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106 CHAPTER 3 MODULAR ARITHMETIC No
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108 CHAPTER 3 MODULAR ARITHMETIC x
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110 CHAPTER 3 MODULAR ARITHMETIC He
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112 CHAPTER 3 MODULAR ARITHMETIC Be
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114 CHAPTER 3 MODULAR ARITHMETIC (c
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116 CHAPTER 3 MODULAR ARITHMETIC (a
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118 CHAPTER 3 MODULAR ARITHMETIC
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120 CHAPTER 3 MODULAR ARITHMETIC mu
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122 CHAPTER 3 MODULAR ARITHMETIC We
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124 CHAPTER 3 MODULAR ARITHMETIC Mo
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128 CHAPTER 3 MODULAR ARITHMETIC Fi
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130 CHAPTER 3 MODULAR ARITHMETIC Fi
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132 CHAPTER 3 MODULAR ARITHMETIC 20
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134 CHAPTER 3 MODULAR ARITHMETIC 1:
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136 CHAPTER 3 MODULAR ARITHMETIC }
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138 CHAPTER 3 MODULAR ARITHMETIC Th
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140 CHAPTER 3 MODULAR ARITHMETIC No
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142 CHAPTER 3 MODULAR ARITHMETIC Th
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146 CHAPTER 3 MODULAR ARITHMETIC E
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148 CHAPTER 3 MODULAR ARITHMETIC th
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150 CHAPTER 3 MODULAR ARITHMETIC 3.
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152 CHAPTER 3 MODULAR ARITHMETIC 3.
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154 CHAPTER 3 MODULAR ARITHMETIC Se
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4 Modular Arithmetic, Decimals, and
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158CHAPTER 4 MODULAR ARITHMETIC, DE
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174 CHAPTER 5 SET THEORY 5.1.1 Defi
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176 CHAPTER 5 SET THEORY (a) Descri
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178 CHAPTER 5 SET THEORY and and Fo
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180 CHAPTER 5 SET THEORY For exampl
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182 CHAPTER 5 SET THEORY (a) ⋂ n
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184 CHAPTER 5 SET THEORY (b) C ∪
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186 CHAPTER 5 SET THEORY Proof. The
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188 CHAPTER 5 SET THEORY (I) Every
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190 CHAPTER 5 SET THEORY Proof. To
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192 CHAPTER 5 SET THEORY Let’stak
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194 CHAPTER 5 SET THEORY 5.4 Hints
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196 CHAPTER 5 SET THEORY Key Formul
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198 CHAPTER 6 FUNCTIONS: BASIC CONC
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264 CHAPTER 7 INTRODUCTION TO CRYPT
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8 Sigma Notation We’re about to s
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300 CHAPTER 8 SIGMA NOTATION And wh
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302 CHAPTER 8 SIGMA NOTATION Applyi
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304 CHAPTER 8 SIGMA NOTATION If we
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306 CHAPTER 8 SIGMA NOTATION Now we
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308 CHAPTER 8 SIGMA NOTATION Exerci
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310 CHAPTER 8 SIGMA NOTATION (c) (d
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314 CHAPTER 8 SIGMA NOTATION where
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316 CHAPTER 8 SIGMA NOTATION 8.5.1
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318 CHAPTER 8 SIGMA NOTATION (a) Le
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320 CHAPTER 8 SIGMA NOTATION (a) We
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322 CHAPTER 8 SIGMA NOTATION We may
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328 CHAPTER 8 SIGMA NOTATION (a) Ex
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330 CHAPTER 8 SIGMA NOTATION notati
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332 CHAPTER 8 SIGMA NOTATION The ne
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334 CHAPTER 8 SIGMA NOTATION Tr(log
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336 CHAPTER 8 SIGMA NOTATION In the
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340 CHAPTER 8 SIGMA NOTATION where
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344 CHAPTER 8 SIGMA NOTATION and we
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346 CHAPTER 8 SIGMA NOTATION (a ×
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348 CHAPTER 8 SIGMA NOTATION This i
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350 CHAPTER 8 SIGMA NOTATION is a r
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352 CHAPTER 8 SIGMA NOTATION Now we
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354 CHAPTER 8 SIGMA NOTATION differ
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356 CHAPTER 8 SIGMA NOTATION • Th
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358 CHAPTER 8 SIGMA NOTATION Pluggi
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360 CHAPTER 8 SIGMA NOTATION 8.8 Hi
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362 CHAPTER 8 SIGMA NOTATION 8.9 St
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364 CHAPTER 8 SIGMA NOTATION 2. 3.
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9 Polynomials In this chapter we’
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368 CHAPTER 9 POLYNOMIALS we find:
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370 CHAPTER 9 POLYNOMIALS According
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372 CHAPTER 9 POLYNOMIALS Remark 9.
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374 CHAPTER 9 POLYNOMIALS Although
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376 CHAPTER 9 POLYNOMIALS This is t
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378 CHAPTER 9 POLYNOMIALS 9.4 More
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380 CHAPTER 9 POLYNOMIALS It turns
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382 CHAPTER 9 POLYNOMIALS c 4 = 4
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384 CHAPTER 9 POLYNOMIALS c 4 = 4
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386 CHAPTER 9 POLYNOMIALS and assoc
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388 CHAPTER 9 POLYNOMIALS ( m ) ∑
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390 CHAPTER 9 POLYNOMIALS Now we mu
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392 CHAPTER 9 POLYNOMIALS 9.6 Polyn
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394 CHAPTER 9 POLYNOMIALS (a) x 2 +
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396 CHAPTER 9 POLYNOMIALS It’s te
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398 CHAPTER 9 POLYNOMIALS (b) Use t
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400 CHAPTER 9 POLYNOMIALS If we set
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402 CHAPTER 9 POLYNOMIALS And here
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404 CHAPTER 9 POLYNOMIALS First ( w
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406 CHAPTER 9 POLYNOMIALS So now we
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408 CHAPTER 9 POLYNOMIALS Since thi
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410 CHAPTER 9 POLYNOMIALS Take a mo
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10 Symmetries of Plane Figures “I
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414 CHAPTER 10 SYMMETRIES OF PLANE
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11 Permutations ”For the real env
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444 CHAPTER 11 PERMUTATIONS But wha
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446 CHAPTER 11 PERMUTATIONS Is μ =
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448 CHAPTER 11 PERMUTATIONS a bijec
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450 CHAPTER 11 PERMUTATIONS (a) Wri
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452 CHAPTER 11 PERMUTATIONS (a) Wri
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454 CHAPTER 11 PERMUTATIONS 11.3.2
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456 CHAPTER 11 PERMUTATIONS Exercis
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458 CHAPTER 11 PERMUTATIONS We may
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460 CHAPTER 11 PERMUTATIONS (ii) In
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462 CHAPTER 11 PERMUTATIONS • 4 d
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464 CHAPTER 11 PERMUTATIONS Then it
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466 CHAPTER 11 PERMUTATIONS (a) S 6
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468 CHAPTER 11 PERMUTATIONS (d) Wha
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470 CHAPTER 11 PERMUTATIONS (a) (12
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472 CHAPTER 11 PERMUTATIONS (a) τ
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474 CHAPTER 11 PERMUTATIONS What Pr
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476 CHAPTER 11 PERMUTATIONS (a) (14
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478 CHAPTER 11 PERMUTATIONS 11.5
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480 CHAPTER 11 PERMUTATIONS the per
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482 CHAPTER 11 PERMUTATIONS Figure
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484 CHAPTER 11 PERMUTATIONS 11.6 Ot
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486 CHAPTER 11 PERMUTATIONS Figure
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488 CHAPTER 11 PERMUTATIONS So far
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490 CHAPTER 11 PERMUTATIONS In ligh
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492 CHAPTER 11 PERMUTATIONS 11.7 Ad
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494 CHAPTER 11 PERMUTATIONS 11.8 Hi
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496 CHAPTER 12 INTRODUCTION TO GROU
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13 Further Topics in Cryptography I
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544 CHAPTER 13 FURTHER TOPICS IN CR
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572CHAPTER 14 EQUIVALENCE RELATIONS
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15 Cosets and Quotient Groups (a.k.
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616CHAPTER 15 COSETS AND QUOTIENT G
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652CHAPTER 16 ERROR-DETECTING AND C
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17 Isomorphisms of Groups Thanks to
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696 CHAPTER 17 ISOMORPHISMS OF GROU
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18 Homomorphisms of Groups In this
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746 CHAPTER 18 HOMOMORPHISMS OF GRO
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19 Group Actions We’ve defined a
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766 CHAPTER 19 GROUP ACTIONS Figure
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768 CHAPTER 19 GROUP ACTIONS So, [(
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770 CHAPTER 19 GROUP ACTIONS Theref
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774 CHAPTER 19 GROUP ACTIONS other
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776 CHAPTER 19 GROUP ACTIONS about
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778 CHAPTER 19 GROUP ACTIONS (c) Re
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780 CHAPTER 19 GROUP ACTIONS where
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782 CHAPTER 19 GROUP ACTIONS rotati
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784 CHAPTER 19 GROUP ACTIONS See Fi
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786 CHAPTER 19 GROUP ACTIONS of rot
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788 CHAPTER 19 GROUP ACTIONS can al
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790 CHAPTER 19 GROUP ACTIONS Let’
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794 CHAPTER 19 GROUP ACTIONS 4. Wha
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796 CHAPTER 19 GROUP ACTIONS Axes c
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798 CHAPTER 19 GROUP ACTIONS Now in
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800 CHAPTER 19 GROUP ACTIONS ♦ Re
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802 CHAPTER 19 GROUP ACTIONS 19.2.7
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804 CHAPTER 19 GROUP ACTIONS groups
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806 CHAPTER 19 GROUP ACTIONS Exampl
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808 CHAPTER 19 GROUP ACTIONS mathem
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810 CHAPTER 19 GROUP ACTIONS b+a+H
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814 CHAPTER 19 GROUP ACTIONS the po
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816 CHAPTER 19 GROUP ACTIONS (b) Sh
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818 CHAPTER 19 GROUP ACTIONS showst
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820 CHAPTER 19 GROUP ACTIONS Proof.
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822 CHAPTER 19 GROUP ACTIONS • St
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824 CHAPTER 19 GROUP ACTIONS (a) Th
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826 CHAPTER 19 GROUP ACTIONS |G| =|
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828 CHAPTER 19 GROUP ACTIONS (a) Cr
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20 Introduction to Rings and Fields
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832 CHAPTER 20 INTRODUCTION TO RING
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21 Appendix: Induction proofs-patte
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900CHAPTER 21 APPENDIX: INDUCTION P
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922 GFDL LICENSE 1. Applicability A
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924 GFDL LICENSE 3. Copying In Quan
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926 GFDL LICENSE If the Modified Ve
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928 GFDL LICENSE If a section in th
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930 GFDL LICENSE
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932 INDEX Caesar, Julius, 265 Cance
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934 INDEX Element centralizer of, 6
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936 INDEX multiplicative, 38 of a c
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938 INDEX definition, 34 Miller-Rab
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Abstract Algebra: Examples and Appl
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