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For a full year introduction to Abstract Algebra at the advanced undergraduate or graduate level. Portions of the book may also be used for various one semester topics courses in algebra. This text is designed to give students insight into the main themes in abstract algebra. 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, students gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. Early introduction to recurring notions such as homomorphisms, isomorphisms, actions and classifications provides a natural, unifying flow for this development.
Introduction to Groups.
Subgroups.
Quotient Groups and Homomorphisms.
Group Actions.
Direct and Semidirect Products and Abelian Groups.
Further Topics in Group Theory.
RING THEORY.
Introduction to Rings.
Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.
Polynomial Rings.
MODULES AND VECTOR SPACES.
Introduction to Module Theory.
Vector Spaces.
Modules over Principal Ideal Domains.
FIELD THEORY AND GALOIS THEORY.
Field Theory.
Galois Theory.
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.
INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS.
Representation Theory and Character Theory.
Examples and Applications of Character Theory.
Appendices.
Index.
David S. Dummit and Richard M. Foote
ISBN13: 978-0471368571For a full year introduction to Abstract Algebra at the advanced undergraduate or graduate level. Portions of the book may also be used for various one semester topics courses in algebra. This text is designed to give students insight into the main themes in abstract algebra. 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, students gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. Early introduction to recurring notions such as homomorphisms, isomorphisms, actions and classifications provides a natural, unifying flow for this development.
Table of Contents
Introduction to Groups.
Subgroups.
Quotient Groups and Homomorphisms.
Group Actions.
Direct and Semidirect Products and Abelian Groups.
Further Topics in Group Theory.
RING THEORY.
Introduction to Rings.
Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.
Polynomial Rings.
MODULES AND VECTOR SPACES.
Introduction to Module Theory.
Vector Spaces.
Modules over Principal Ideal Domains.
FIELD THEORY AND GALOIS THEORY.
Field Theory.
Galois Theory.
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.
INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS.
Representation Theory and Character Theory.
Examples and Applications of Character Theory.
Appendices.
Index.