Elementary Abstract Algebra- Examples and Applications, 2019a

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3 Portions c○2006-2009 by Dave Witte Morris and Joy Morris. Some rights reserved. Permission is granted to copy, distribute and/or modify this document under the terms of 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. The license may be found at: https://www.gnu.org/licenses/fdl-1.3.en.html ISBN: 978-0-359-04211-1
Contents Forward 1 To the student: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 To the instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Additional resources . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Organization Plan of the Book 6 Glossary of Symbols 10 1 Preliminaries 12 1.1 In the Beginning . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Integers, rational numbers, real numbers . . . . . . . . . . . . 13 1.2.1 Properties of arithmetic operations . . . . . . . . . . . 14 1.2.2 Order relations . . . . . . . . . . . . . . . . . . . . . . 17 1.2.3 Manipulating equations and inequalities . . . . . . . . 18 1.2.4 Exponentiation (VERY important) . . . . . . . . . . . 22 1.3 Test yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4
Page 1 and 2: Abstract Algebra: Examples and Appl
Page 3: 2 Material from the 2012 version of
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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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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784 CHAPTER 19 GROUP ACTIONS See Fi
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786 CHAPTER 19 GROUP ACTIONS of rot
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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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