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by David S. Dummit and Richard M. Foote

Edition: 2ND 99Copyright: 1999

Publisher: Prentice Hall, Inc.

Published: 1999

International: No

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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.

- NEW--Builds on an already widely acclaimed text through the addition of substantial new material in areas that include: tensor products, commutative rings, algebraic number theory and introductory algebraic geometry.
- NEW--Material in the first edition will also be enhanced with some new examples and exercises.
- The exposition and development make the text accessible to undergraduates, yet its scope and depth also make it ideal for courses at the graduate level.
- 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 (e.g. Berlekamp's Algorithm for factoring a polynomial mod p).
- Contains many topics not usually found in introductory texts. These are made accessible through clear exposition and insightful examples and students are able to see how they fit naturally into the main themes of algebra. Examples of this include:
- Rings of algebraic integers. Semidirect products and splitting of extensions. Criteria for the solvability of a quintic. Dedekind Domains.
- The structure of the book permits instructors and students to pursue certain areas (group theory, Galois theory, etc.) from their beginnings through to an in-depth treatment, or to survey a wider range of areas, seeing how various themes recur and how different structures are related.

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.

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Summary

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.

- NEW--Builds on an already widely acclaimed text through the addition of substantial new material in areas that include: tensor products, commutative rings, algebraic number theory and introductory algebraic geometry.
- NEW--Material in the first edition will also be enhanced with some new examples and exercises.
- The exposition and development make the text accessible to undergraduates, yet its scope and depth also make it ideal for courses at the graduate level.
- 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 (e.g. Berlekamp's Algorithm for factoring a polynomial mod p).
- Contains many topics not usually found in introductory texts. These are made accessible through clear exposition and insightful examples and students are able to see how they fit naturally into the main themes of algebra. Examples of this include:
- Rings of algebraic integers. Semidirect products and splitting of extensions. Criteria for the solvability of a quintic. Dedekind Domains.
- The structure of the book permits instructors and students to pursue certain areas (group theory, Galois theory, etc.) from their beginnings through to an in-depth treatment, or to survey a wider range of areas, seeing how various themes recur and how different structures are related.

Table of Contents

GROUP THEORY.

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.

Publisher Info

Publisher: Prentice Hall, Inc.

Published: 1999

International: No

Published: 1999

International: No

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.

- NEW--Builds on an already widely acclaimed text through the addition of substantial new material in areas that include: tensor products, commutative rings, algebraic number theory and introductory algebraic geometry.
- NEW--Material in the first edition will also be enhanced with some new examples and exercises.
- The exposition and development make the text accessible to undergraduates, yet its scope and depth also make it ideal for courses at the graduate level.
- 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 (e.g. Berlekamp's Algorithm for factoring a polynomial mod p).
- Contains many topics not usually found in introductory texts. These are made accessible through clear exposition and insightful examples and students are able to see how they fit naturally into the main themes of algebra. Examples of this include:
- Rings of algebraic integers. Semidirect products and splitting of extensions. Criteria for the solvability of a quintic. Dedekind Domains.
- The structure of the book permits instructors and students to pursue certain areas (group theory, Galois theory, etc.) from their beginnings through to an in-depth treatment, or to survey a wider range of areas, seeing how various themes recur and how different structures are related.

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.