Lectures on Abstract Algebra - Softcover

Inhaltsangabe

This book is primarily intended for undergraduate students in Mathematics, pursuing courses that feature abstract algebra. All aspects of abstract algebra that you would expect to encounter in an undergraduate programme of study are covered, including ring theory, group theory and the beginnings of Galois theory. Unusually for an abstract algebra text, five chapters on linear algebra are also included, making the text a self-contained introduction to undergraduate algebra. The book will be of use throughout your undergraduate studies, and beyond.Very little is assumed by way of previous study. It is assumed only that you are familiar with basic topics in algebra, such as complex numbers, matrices, and solving systems of linear equations. From these beginnings, you will be led step by step into the study of various algebraic structures, including groups, vector spaces and rings, and structure-preserving mappings between such objects such as homomorphisms of groups and linear transformations of vector spaces. Important theorems and results are presented and discussed at each stage, with proofs that will enable you to understand the logical development of the subject. Mathematical proof is an aspect of the subject which many students find challenging, and care has been taken to ensure that proofs are easy to follow with all steps clearly explained.In the last six chapters, you will be introduced to more novel topics that go beyond the typical undergraduate curriculum. These include exterior algebras, matrix groups and their associated Lie algebras, normed real algebras and Clifford algebras. Any of these topics could form the basis for a final year undergraduate dissertation, or provide a gentle introduction to graduate study in algebra.Throughout the text, the emphasis is upon presenting topics in an accessible manner, with worked examples which will help you to build a firm understanding of each new concept as it is introduced. Each chapter concludes with a comprehensive set of exercises designed to test and consolidate your understanding of the ideas presented. Solutions to the exercises are given at the end of the book.The book is based on lectures given by Michael Butler at the University of Bolton in the U.K. between 1994 and 2019. These were attended by students from an exceptionally wide range of backgrounds. The lecture notes that this book grew from were refined in the light of teaching and in response to feedback from students, over many deliveries of the material. Dr Butler has extensive experience of teaching abstract algebra to undergraduates, and his passion for teaching the subject has shaped this book. CONTENTS: 1 The Group Axioms and Examples 2 Subgroups and Group Homomorphisms 3 Vector Spaces 4 Subspaces and Linear Transformations 5 The Basis for a Vector Space 6 Eigenvalues and Eigenvectors 7 Inner Product Spaces 8 Cosets and Quotient Groups 9 Group Actions 10 Simple Groups 11 Soluble Groups 12 Fields and their Extensions 13 The Galois Group 14 The Ring Axioms and Examples15 Subrings, Ideals and Ring Homomorphisms 16 Quotient Rings 17 Integral Domains and Fields 18 Finite Fields 19 Factorisation in an Integral Domain 20 Vector Spaces with Products 21 The Exterior Algebra of a Vector Space 22 Lie Algebras 23 Matrix Groups and their Tangent Spaces 24 Normed Real Algebras 25 Tensor Products and Clifford Algebras.

Die Inhaltsangabe kann sich auf eine andere Ausgabe dieses Titels beziehen.