Sivaramakrishnan R. Certain Number-Theoretic Episodes In Algebra

2nd. ed. - Boca Raton: CRC Press, 2019. - 443 p. - ISBN: 1138495786.This is the second edition of the text: CERTAIN NUMBER-THEORETIC EPISODES IN ALGEBRA published by Chapman & Hall/CRC (Taylor & Francis Group) Boca Raton, FL 33487–2742 in 2006. A noteworthy feature of this edition is that the book has been completely revised and updated. The purpose of the book is to describe some of the various situations in number theory which are capable of interpretation via algebraic proofs and to point out the contexts where analogues of known number-theoretic results, such as:The fundamental theorem of arithmetic.
The Chinese Remainder Theorem.
Quadratic reciprocity law and,
The Goldbach problem.are available in algebra. The book is intended for the use of senior undergraduates who aspire for a research career with possible specialization in number theory, combinatorial mathematics or algebra.About the Author.
Chapter-Wise Description of the Contents.
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 (mod r) (r > 1).
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.
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.
Glimpses of the Theory of Algebraic Numbers
Dedekind Domains.
Algebraic Number Fields.
Some Additional Topics
Vaidyanathaswamy’s Class-Division of Integers Modulo r.
Burnside’s Lemma and a Few of Its Applications.
On Cyclic Codes of Length n over F.
An Analogue of the Goldbach Problem.
App. A. On the Partition Function p(r) (r > 1).
App. B. Thumb-Nail Sketches of Biographies of Forty-One Prominent Mathematicians.
App. C. Suggested for Further Study / Reading.
List of symbols.
Author Index.
Index of Mathematical Terms.True PDF