Chapman and Hall/CRC
412 pages | 47 B/W Illus.
The book attempts to point out the interconnections between number theory and algebra with a view to making a student understand certain basic concepts in the two areas forming the subject-matter of the book.
Section A - ELEMENTS OF THE THEORY OF NUMBERS. From Euclid to Lucas: Elementary theorems revisited. Solutions of Congruences, Primitive Roots. The Chinese Remainder Theorem. M¨obius 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 class-division 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.