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 Sivaramakrishnan, R., 1936 author.
 Second edition.  Boca Raton : CRC Press, Taylor & Francis Group, [2019]
 Description
 Book — xxxi, 411 pages : illustrations ; 25 cm
 Summary

 Section A  ELEMENTS OF THE THEORY OF NUMBERS. From Euclid to Lucas: Elementary theorems revisited. Solutions of Congruences, Primitive Roots. The Chinese Remainder Theorem. Mobius inversion. Quadratic Residues. Decomposition of a number as a sum of two or four squares. Dirichlet Algebra of Arithmetical Functions. Modular arithmetical functions. A generalization of Ramanujan sums. Ramanujan expansions of multiplicative arithmetic functions. Section B  SELECTED TOPICS IN ALGEBRA. On the uniqueness of a group of order r (r > 1). Quadratic Reciprocity in a finite group. Commutative rings with unity. Noetherian and Artinian rings. Section C  GLIMPSES OF THE THEORY OF ALGEBRAIC NUMBERS. Dedekind domains. Algebraic number fields. Section D  SOME ADDITIONAL TOPICS. Vaidyanathaswamy's classdivision of integers modulo r. Burnside's lemma and a few of its applications. On cyclic codes of length n over Fq. An Analogue of the Goldbach problem. Appendix A. Appendix B. Appendix C. Index.
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QA247 .S5725 2019  Unknown 
 Sivaramakrishnan, R., 1936
 Boca Raton, FL : Chapman & Hall/CRC, c2007.
 Description
 Book — 632 p. ; 25 cm.
 Summary

 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 Workedout 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 Workedout examples Euclidean Domains Historical perspective Introduction Z as a Euclidean domain Quadratic number fields Almost Euclidean domains Notes with illustrative examples Workedout 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 Workedout 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 Workedout 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 Workedout 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 workedout 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 JordanHolder theorem Noncommutative rings Boolean algebras Notes with illustrative examples Workedout examples The Role of the Mobius FunctionAbstract Mobius Inversion Historical perspective Introduction Abstract Mobius inversion Incidence algebra of n A n matrices Vector spaces over a finite field Notes with illustrative examples Workedout 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 Workedout examples Catalan numbers Semigroups and Certain Convolution Algebras Historical perspective Introduction Semigroups Semicharacters Finite dimensional convolution algebras Abstract arithmetical functions Convolutions in general A functionaltheoretic algebra Notes with illustrative examples Workedout examples A GLIMPSE OF ALGEBRAIC NUMBER THEORY Noetherian and Dedekind Domains Historical perspective Introduction Noetherian rings More about ideals Jacobson radical The LaskerNoether decomposition theorem Dedekind domains The Chinese remainder theorem revisited Integral domains having finite norm property Notes with illustrative examples Workedout examples Algebraic Number Fields Historical perspective Introduction The ideal class group Cyclotomic fields Halffactorial domains The Pell equation The Cakravala method Dirichlet's unit theorem Notes with illustrative examples Formally real fields Workedout 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 normpreserving transformations Notes with illustrative examples Workedout 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 Workedout 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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SAL3 (offcampus storage)
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QA247 .S5725 2007  Available 
3. Classical theory of arithmetic functions [1989]
 Sivaramakrishnan, R., 1936
 New York : M. Dekker, c1989.
 Description
 Book — xii, 386 p. ; 24 cm.
 Summary

This volume focuses on the classical theory of numbertheoretic functions emphasizing algebraic and multiplicative techniques. It contains many structure theorems basic to the study of arithmetic functions, including several previously unpublished proofs.
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Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

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QA245 .S59 1989  Unknown 
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