An Introduction to Abstract Algebra
The principal structures of modern algebra and its applications form the main topic of the book. In twelve chapters, the author explains the main topics in abstract algebra with many useful examples.
The first two chapters introduce fundamental concepts (set, class, properties of integers, GCD, Euclidean algorithm, the fundamental theorem of arithmetics). The concept of a group is explored in Chapters 3, 4, 5 and 9. The exposition follows a historical development, and the author uses the group of permutations as a motivating example. Lagrange’s theorem, the isomorphism theorems and other classic facts concerning normal and quotient groups are included as a standard part of a university algebra course. Group actions are used to investigate the group structure. Algebraic structures with two binary operations are introduced in Chapter 6: rings, Euclidean domains, roots of polynomials, and splitting fields. Vector spaces are used in many applications: they are discussed in a separate chapter with many examples. Chapters 10 and 11 contain a description of fields and Galois theory, with applications to orthogonal latin squares, Steiner systems and the solvability of equations by radicals. The final chapter contains some additional material useful for applications.
This book will find its readers among specialists in mathematics, physics and informatics, who need to acquire a basic knowledge of algebra and its applications. The book should be useful for university students attending algebra courses.