Graduate Texts in Mathematics 4

Hemant Kumar Pathak

Abstract Algebra and Applications

This textbook discusses all useful topics in abstract algebra and applications. It contains complete definitions of topics well-explained by suitable examples, explanations, and proofs throughout the book. The initial chapters cover the topics such as groups, subgroups, cyclic groups, permutation groups, quotient groups, normal series, structure theorems of groups, ring theory and polynomials, vector spaces and linear mappings usually taught at the undergraduate level. The later chapters contain some additional topics such as field extensions. Galois theory, modules, Noetherian and Artinian modules, and rings. Linear transformations, associated algebra and canonical forms, Smith normal form over a PID and rank, finitely generated modules over a PID which are taught over a period of four decades and can serve as a textbook on Abstract Algebra and Applications.

This book is designed as a textbook for a two-semester course in abstract algebra for upper undergraduate and beginning graduate students. The book is also accessible to junior mathematics majors who have studied set theory, elementary calculus, and number systems. The essential pre-requisites for reading this book are a basic understanding of fundamental notions of set theory and some knowledge of number systems. The reader should know some basic facts about calculus of one, two and multivariable.