About the Book :
Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings.
Interesting Facts :
Over 1500 exercises, many with multiple parts, ranging in scope from routine to fairly sophisticated, and ranging in purpose from basic application of text material to exploration of important theoretical or computational techniques. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible. Contains many topics not usually found in a basic algebra book such as rings of algebraic integers, semidirect products and the theory of extensions, criteria for Principal Ideal Domains, criteria for solvability of a quintic, and Dedekind Domains.
PART I: GROUP THEORY.Introduction to Groups.Subgroups.Quotient Group and Homomorphisms.Group Actions.Direct and Semidirect Products and Abelian Groups.Further Topics in Group Theory.PART II: RING THEORY.Introduction to Rings.Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.Polynomial Rings.PART III: MODULES AND VECTOR SPACES.Introduction to Module Theory.Vector Spaces.Modules over Principal Ideal Domains.PART IV: FIELD THEORY AND GALOIS THEORY.Field Theory.Galois Theory.PART V: AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOMETRY, AND HOMOLOGICAL ALGEBRA.Commutative Rings and Algebraic Geometry.Artinian Rings, Discrete Valuation Rings, and Dedekind Domains.Introduction to Homological Algebra and Group Cohomology.PART VI: INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS.Representation Theory and Character Theory.Examples and Applications of Character Theory.Appendix I: Cartesian Products and Zorn's Lemma.Appendix II: Category Theory.Index.
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