Certain Number-Theoretic Episodes in Algebra
This book is focused on some important aspects of interrelations between number theory and commutative algebra. The book is divided into four parts; each chapter starts with a historical overview and closes with illustrative examples. Part I deals with elements of number theory and algebra. The author discusses many interesting topics in detail, such as Euler's theorem, Fermat's theorem, Lagrange's theorem, integral domains of rational integers, Euclidean domains, rings of polynomials and formal power series, the Chinese remainder theorem, the reciprocity law and finite groups. Part II explains certain aspects of algebraic structures (including ordered fields, fields with valuation, abstract Möbius inversion, generating functions, finite semigroups and convolutions algebras). Part III gives an overview of foundations of algebraic number theory. Noetherian and Dedekind domains, Pell's equations and their solutions, the Dirichlet unit theorem and the ideal class-group are presented in detail. Part IV describes some other interrelations between algebraic number theory and abstract algebra (rings of arithmetic functions, the polynomial analogue of the Goldbach problem, finite dimensional algebras and algebraic structures). Since the reader needs only a rudimentary knowledge of elementary number theory and algebra, the book can be recommended to anyone interested in these domains.