A Course on Abstract Algebra

This textbook provides an introduction to abstract algebra for advanced undergraduate students. Based on the authors' lecture notes at the Department of Mathematics, National Chung Cheng University of Taiwan, it begins with a description of the algebraic structures of the ring and field of rational numbers. Abstract groups are then introduced. Technical results such as Lagrange's Theorem and Sylow's Theorems follow as applications of group theory. Ring theory forms the second part of abstract algebra, with the ring of polynomials and the matrix ring as basic examples. The general theory of ideals as well as maximal ideals in the rings of polynomials over the rational numbers are also discussed. The final part of the book focuses on field theory, field extensions and then Galois theory to illustrate the correspondence between the Galois groups and field extensions. This textbook is more accessible and less ambitious than most existing books covering the same subject. Readers will also find the pedagogical material very useful in enhancing the teaching and learning of abstract algebra. Contents: Algebraic Structure of Numbers Basic Notions of Groups Cyclic Groups Permutation Groups Counting Theorems Group Homomorphisms The Quotient Group Finite Abelian Groups Sylow Theorems and Applications Introduction to Group Presentations Types of Rings Ideals and Quotient Rings Ring Hormomorphisms Polynomial Rings Factorization Vector Spaces Over an Arbitrary Field Field Extensions All About Roots Galois Pairing Applications of the Galois Paring Readership: Advanced undergraduates and academics in pure mathematics. Key Features: Does not require any special background in algebra, thus making it easy for students to learn and lecturers to teach Includes exercises after each section to enhance comprehension Provides a clearer picture of groups, rings and fields than existing books in the same area Keywords: Abstract Algebra; Group Theory; Sylow's Theorems; Ring Theory; Field Theory