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 ELEMENTS OF NUMBER THEORY AND ALGEBRA  
Theorems of Euler, Fermat and Lagrange  
Historical perspective  
Introduction  
The quotient ring Z / rZ  
An elementary counting principle  
Fermat’s two squares theorem   
Lagrange’s four squares theorem   
Diophantine equations   
Notes with illustrative examples  
Worked-out examples 
The Integral Domain of Rational Integers  
Historical perspective  
Introduction  
Ordered integral domains  
Ideals in a commutative ring  
Irreducibles and primes  
GCD domains  
Notes with illustrative examples  
Worked-out examples 
Euclidean Domains   
Historical perspective  
Introduction  
Z as a Euclidean domain  
Quadratic number fields  
Almost Euclidean domains  
Notes with illustrative examples   
Worked-out examples 
Rings of Polynomials and Formal Power Series   
Historical perspective  
Introduction   
Polynomial rings   
Elementary arithmetic functions  
Polynomials in several indeterminates   
Ring of formal power series  
Finite fields and irreducible polynomials   
More about irreducible polynomials   
Notes with illustrative examples  
Worked-out examples 
The Chinese Remainder Theorem and the Evaluation of Number of Solutions of a Linear Congruence with Side Conditions   
Historical perspective  
Introduction  
The Chinese Remainder theorem   
Direct products and direct sums  
Even functions (mod r)   
Linear congruences with side conditions   
The Rademacher formula   
Notes with illustrative examples   
Worked-out examples 
Reciprocity Laws   
Historical perspective  
Introduction  
Preliminaries   
Gauss lemma  
Finite fields and quadratic reciprocity law  
Cubic residues (mod p)   
Group characters and the cubic reciprocity law  
Notes with illustrative examples  
A comment by W. C. Waterhouse  
Worked-out examples  
Finite Groups  
Historical perspective  
Introduction  
Conjugate classes of elements in a group  
Counting certain special representations of a group element  
Number of cyclic subgroups of a finite group  
A criterion for the uniqueness of a cyclic group of order r  
Notes with illustrative examples   
A worked-out example   
An example from quadratic residues 
THE RELEVANCE OF ALGEBRAIC STRUCTURES TO  
NUMBER THEORY  
Ordered Fields, Fields with Valuation and Other Algebraic Structures  
Historical perspective  
Introduction  
Ordered fields   
Valuation rings   
Fields with valuation  
Normed division domains  
Modular lattices and Jordan-Holder theorem  
Non-commutative rings   
Boolean algebras   
Notes with illustrative examples  
Worked-out examples 
The Role of the Mobius Function?Abstract Mobius Inversion   
Historical perspective  
Introduction  
Abstract Mobius inversion  
Incidence algebra of n × n matrices   
Vector spaces over a finite field  
Notes with illustrative examples  
Worked-out examples  
The Role of Generating Functions  
Historical perspective  
Introduction  
Euler’s theorems on partitions of an integer   
Elliptic functions  
Stirling numbers and Bernoulli numbers  
Binomial posets and generating functions   
Dirichlet series   
Notes with illustrative examples   
Worked-out examples   
Catalan numbers 
Semigroups and Certain Convolution Algebras   
Historical perspective  
Introduction   
Semigroups  
Semicharacters   
Finite dimensional convolution algebras  
Abstract arithmetical functions   
Convolutions in general  
A functional-theoretic algebra  
Notes with illustrative examples  
Worked-out examples 
A GLIMPSE OF ALGEBRAIC NUMBER THEORY  
Noetherian and Dedekind Domains   
Historical perspective   
Introduction  
Noetherian rings   
More about ideals  
Jacobson radical  
The Lasker-Noether decomposition theorem   
Dedekind domains   
The Chinese remainder theorem revisited  
Integral domains having finite norm property   
Notes with illustrative examples   
Worked-out examples  
Algebraic Number Fields   
Historical perspective  
Introduction  
The ideal class group  
Cyclotomic fields   
Half-factorial domains  
The Pell equation  
The Cakravala method   
Dirichlet’s unit theorem   
Notes with illustrative examples  
Formally real fields   
Worked-out examples 
SOME MORE INTERCONNECTIONS  
Rings of Arithmetic Functions   
Historical perspective   
Introduction   
Cauchy composition (mod r)   
The algebra of even functions (mod r)   
Carlitz conjecture  
More about zero divisors  
Certain norm-preserving transformations   
Notes with illustrative examples  
Worked-out examples 
Analogues of the Goldbach Problem  
Historical perspective  
Introduction   
The Riemann hypothesis   
A finite analogue of the Goldbach problem  
The Goldbach problem in Mn(Z)   
An analogue of Goldbach theorem via polynomials over finite fields   
Notes with illustrative examples  
A variant of Goldbach conjecture 
An Epilogue: More Interconnections   
Introduction  
On commutative rings   
Commutative rings without maximal ideals  
Infinitude of primes in a PID   
On the group of units of a commutative ring  
Quadratic reciprocity in a finite group  
Worked-out examples 
True/False Statements: Answer Key  
Index of Some Selected Structure Theorems/Results   
Index of Symbols and Notations   
Bibliography  
Subject Index  
Index of names 
Each chapter includes exercises and references.

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