Certain Number-Theoretic Episodes In Algebra, Second Edition (Chapman & Hall/CRC Pure and Applied Mathematics)

444 Pages·2019·12.661 MB·English

by  R Sivaramakrishnan

Certain Number-Theoretic
Episodes in Algebra
Second Edition

Certain Number-Theoretic
Episodes in Algebra
Second Edition
Sivaramakrishnan. R
CRC Press
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Dedicatedtothememoryofmybelovedbrother
RamakrishnaiyerRaman(1933–2015)
whowasamanofdeepconvictions,committedtohardworkand
unwaveringpursuitofprofessionalgoals. Imisshisguidance.
Sivaramakrishnan. R

CONTENTS
Preface................................................. ..xv
Acknowledgment....................................... ..xxi
AbouttheAuthor ..................................... ..xxiii
Chapter-WiseDescriptionoftheContents............... ..xxv
SectionI-ELEMENTSOFTHETHEORYOFNUMBERS..1
1.FromEuclidtoLucas: ElementaryTheoremsRevisited.. ..3
Introduction............................................ ..3
1.1. ThequotientringZ/rZ (r > 1)..................... ..6
1.2. Congruencesmoduloaprime...................... ..11
1.3. Fermat’stwo-squarestheorem..................... ..16
1.4. Lagrange’sfour-squarestheorem .................. ..19
1.5. Worked-outexamples ............................ ..22
/
1.6. Notes Remarks ................................. ..26
Exercises........................................... ..29
References.......................................... ..30
2.SolutionsofCongruences,PrimitiveRoots............. ..33
Introduction........................................... ..33
2.1. Theoremsoncongruences......................... ..34
2.2. Worked-outexamples ............................ ..37
/
2.3. Notes Remarks ................................. ..38
Exercises........................................... ..39
References.......................................... ..40
3.TheChineseRemainderTheorem..................... ..41
3.1. Introduction..................................... ..41
3.2. TheChineseRemainderTheorem.................. ..44
3.3. Worked-outexamples ............................ ..47
/
3.4. Notes Remarks ................................. ..48
vii
viii
Exercises........................................... ..49
References.......................................... ..52
4.Mo¨biusInversion..................................... ..53
Introduction........................................... ..53
4.1. AbstractMo¨biusinversion ........................ ..54
4.2. Deduction: Mo¨biusinversionofnumbertheory ..... ..58
4.3. ThepowersetP(X)ofafiniteset X ................ ..60
4.4. Aworked-outexample ........................... ..62
/
4.5. Notes Remarks ................................. ..63
Exercises........................................... ..64
References.......................................... ..68
5.QuadraticResidues(modr)(r > 1) .................... ..69
Introduction........................................... ..69
5.1. Preliminaries: Gauss’lemma...................... ..70
5.2. Eisensteinlemma ................................ ..72
5.3. Quadraticreciprocitylaw......................... ..75
5.4. FirstSupplementtoquadraticreciprocitylaw....... ..75
5.5. Secondsupplementtoquadraticreciprocitylaw..... ..76
5.6. TheJacobisymbol............................... ..76
5.7. Worked-outexamples ............................ ..77
/
5.8. Notes Remarks ................................. ..79
Exercises........................................... ..80
References.......................................... ..81
6. Decomposition of a Number as a Sum of Two or Four
Squares .............................................. ..83
Introduction........................................... ..83
6.1. Gaussianintegers ................................ ..86
6.2. Integralquaternions.............................. ..87
6.3. Landau’sTheorem ............................... ..90
6.4. Worked-outexamples ............................ ..90
/
6.5. Notes Remarks ................................. ..92
Exercises........................................... ..93
References.......................................... ..94
7.DirichletAlgebraofArithmeticalFunctions ........... ..95
Introduction........................................... ..95
ix
7.1. Arithmeticalconvolutions......................... ..96
7.2. Arithmeticfunctions ............................. ..97
7.3. Mo¨biusinversion(anotherform)................... ..98
7.4. Unitaryconvolution............................. ..100
7.5. UFDpropertyoftheringofarithmeticfunctions... ..101
7.6. Worked-outexamples ........................... ..105
/
7.7. Notes Remarks................................ ..108
Exercises.......................................... ..110
References ........................................ ..112
8.ModularArithmeticalFunctions ..................... ..115
Introduction.......................................... ..115
8.1. Eckford Cohen’s orthogonal property for Ramanujan
sums.............................................. ..117
8.2. FiniteFourierseriesrepresentationsofevenfunctions
(modr)............................................ ..121
8.3. Anapplication.................................. ..124
8.4. Aworked-outexample .......................... ..125
/
8.5. Notes Remarks................................ ..126
Exercises.......................................... ..128
References ........................................ ..129
9.AGeneralizationofRamanujanSums................ ..131
Introduction.......................................... ..131
9.1. Jordan’stotient J (r) ............................ ..132
k
9.2. Residuesystems(modk,r)....................... ..133
9.3. AgeneralizationofC(n,r)....................... ..134
9.4. Anapplication.................................. ..137
9.5. Worked-outexamples ........................... ..138
/
9.6. Notes Remarks................................ ..140
Exercises.......................................... ..140
References ........................................ ..143
10. Ramanujan Expansions of Multiplicative Arithmetic
Functions ........................................... ..145
Introduction.......................................... ..145
10.1. Averagesofevenfunctions(modr).............. ..148
10.2. Seriesexpansions.............................. ..151
10.3. Worked-outexamples.......................... ..152