A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic

721 Pages·2010·3.689 MB·English

by  Jeffrey Bergen

A Concrete Approach to
Abstract Algebra
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A Concrete Approach to
Abstract Algebra
From the Integers to the Insolvability of the Quintic
Jeffrey Bergen
DePaulUniversity
Chicago,Illinois
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Bergen,Jeffrey,1955.
Aconcreteapproachtoabstractalgebra:fromtheintegerstotheinsolvabilityofthequintic/JeffreyBergen.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-12-374941-3(hardcover:alk.paper)1.Algebra,Abstract.I.Title.
QA162.B452010
512’.02–dc22
2009035349
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To Donna
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Contents
Preface...............................................................................xi
AUser’sGuide...................................................................... xv
Acknowledgments ...................................................................xix
Chapter1 WhatThisBookIsaboutandWhoThisBookIsfor.......................1
1.1 Algebra...........................................................................2
1.1.1 FindingRootsofPolynomials...................................................2
1.1.2 ExistenceofRootsofPolynomials..............................................4
1.1.3 SolvingLinearEquations........................................................5
1.2 Geometry ........................................................................6
1.2.1 RulerandCompassConstructions...............................................6
1.3 Trigonometry ....................................................................7
1.3.1 RationalValuesofTrigonometricFunctions....................................7
1.4 Precalculus.......................................................................8
1.4.1 RecognizingPolynomialsUsingData...........................................8
1.5 Calculus.........................................................................10
1.5.1 PartialFractionDecomposition................................................ 10
1.5.2 DetectingMultipleRootsofPolynomials..................................... 12
ExercisesforChapter1 ........................................................14
Chapter2 ProofandIntuition.......................................................19
2.1 TheWellOrderingPrinciple...................................................20
2.2 ProofbyContradiction.........................................................26
2.3 MathematicalInduction........................................................29
MathematicalInduction—FirstVersion.......................................30
MathematicalInduction—FirstVersionRevisited............................32
MathematicalInduction—SecondVersion....................................37
ExercisesforSections2.1,2.2,and2.3 .......................................37
2.4 FunctionsandBinaryOperations..............................................46
ExercisesforSection2.4.......................................................56
vii
Contents
Chapter3 TheIntegers .............................................................61
3.1 PrimeNumbers.................................................................61
3.2 UniqueFactorization...........................................................64
3.3 DivisionAlgorithm.............................................................67
ExercisesforSections3.1,3.2,and3.3 .......................................71
3.4 GreatestCommonDivisors....................................................76
3.5 EuclideanAlgorithm ...........................................................79
ExercisesforSections3.4and3.5.............................................91
Chapter4 TheRationalNumbersandtheRealNumbers............................97
4.1 RationalNumbers ..............................................................97
4.2 IntermediateValueTheorem................................................. 105
ExercisesforSections4.1and4.2........................................... 113
4.3 EquivalenceRelations........................................................ 118
ExercisesforSection4.3..................................................... 128
Chapter5 TheComplexNumbers................................................. 137
5.1 ComplexNumbers ........................................................... 137
5.2 FieldsandCommutativeRings.............................................. 140
ExercisesforSections5.1and5.2........................................... 148
5.3 ComplexConjugation........................................................ 154
5.4 AutomorphismsandRootsofPolynomials ................................. 163
ExercisesforSections5.3and5.4........................................... 169
5.5 GroupsofAutomorphismsofCommutativeRings......................... 177
ExercisesforSection5.5..................................................... 182
Chapter6 TheFundamentalTheoremofAlgebra ................................. 189
6.1 RepresentingRealNumbersandComplexNumbersGeometrically ...... 189
6.2 RectangularandPolarForm ................................................. 199
ExercisesforSections6.1and6.2........................................... 203
6.3 Demoivre’sTheoremandRootsofComplexNumbers .................... 208
6.4 AProofoftheFundamentalTheoremofAlgebra .......................... 215
ExercisesforSections6.3and6.4........................................... 222
Chapter7 TheIntegersModulon ................................................. 227
7.1 DefinitionsandBasicProperties............................................. 227
7.2 ZeroDivisorsandInvertibleElements...................................... 233
ExercisesforSections7.1and7.2........................................... 241
7.3 TheEulerφ Function......................................................... 248
7.4 PolynomialswithCoefficientsinZ ........................................ 256
n
ExercisesforSections7.3and7.4........................................... 260
viii
Contents
Chapter8 GroupTheory.......................................................... 265
8.1 DefinitionsandExamples.................................................... 265
I.CommutativeRingsandFieldsunderAddition ........................ 266
II.InvertibleElementsinCommutativeRingsunderMultiplication ..... 266
III.BijectionsofSets......................................................... 267
ExercisesforSection8.1..................................................... 288
8.2 TheoremsofLagrangeandSylow........................................... 294
ExercisesforSection8.2..................................................... 318
8.3 SolvableGroups.............................................................. 322
ExercisesforSection8.3..................................................... 342
8.4 SymmetricGroups ........................................................... 347
ExercisesforSection8.4..................................................... 361
Chapter9 PolynomialsovertheIntegersandRationals............................ 365
9.1 IntegralDomainsandHomomorphismsofRings .......................... 365
ExercisesforSection9.1..................................................... 374
9.2 RationalRootTestandIrreduciblePolynomials............................ 379
ExercisesforSection9.2..................................................... 387
9.3 Gauss’LemmaandEisenstein’sCriterion .................................. 390
ExercisesforSection9.3..................................................... 397
9.4 ReductionModulop.......................................................... 398
ExercisesforSection9.4..................................................... 408
Chapter10 RootsofPolynomialsofDegreeLessthan5 .......................... 411
10.1 FindingRootsofPolynomialsofSmallDegree ............................ 411
10.2 ABriefLookatSomeConsequencesofGalois’Work ..................... 418
ExercisesforSections10.1and10.2........................................ 420
Chapter11 RationalValuesofTrigonometricFunctions........................... 423
11.1 ValuesofTrigonometricFunctions.......................................... 424
ExercisesforSection11.1 ................................................... 433
Chapter12 PolynomialsoverArbitraryFields..................................... 437
12.1 SimilaritiesbetweenPolynomialsandIntegers............................. 437
12.2 DivisionAlgorithm........................................................... 444
ExercisesforSections12.1and12.2........................................ 453
12.3 IrreducibleandMinimumPolynomials ..................................... 457
12.4 EuclideanAlgorithmandGreatestCommonDivisors...................... 460
ExercisesforSections12.3and12.4........................................ 470
12.5 FormalDerivativesandMultipleRoots..................................... 474
ExercisesforSection12.5 ................................................... 484
ix