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### Abstract Algebra David S. Dummit and Richard M. Foote Solution Manual by Bryan F´elix

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Abstract Algebra David S. Dummit and Richard M. Foote Solution Manual written by Bryan F´elix 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 student's understanding. The text book is designed for a full-year introduction to abstract algebra at the advanced undergraduate or graduate level, but contains substantially more material than would normally be covered in one year. Portions of the book may also be used for various one-semester topics courses in advanced algebra, each of which would provide a solid background for a follow-up course delving more deeply into one of many possible areas: algebraic number theory, algebraic topology, algebraic geometry, representation theory, Lie groups, group theory, ring theory etc. . This is an other great mathematics book cover the following topics.

• 0. Preliminaries

• 0.1 Basics
0.2 Properties of the Integers
0.3 Z In Z : The Integers Modulon
• Part 1 - GROUP THEORY

• 1. Introduction to Groups

1.1 Basic Axioms and Examples
1.2 Dihedral Groups
1.3 Symmetric Groups
1.4 Matrix Groups
1.5 The Quaternion Group
1.6 Homomorphisms and Isomorphisms
1.7 Group Actions

• 2. Subgroups

2.1 Definition and Examples
2.2 Centralizers and Normalizers, Stabilizers and Kernels
2.3 Cyclic Groups and Cyclic Subgroups
2.4 Subgroups Generated by Subsets of a Group
2.5 The Lattice of Subgroups of a Group

• 3. Quotient Groups and Homomorphisms

3.1 Definitions and Examples
3.2 More on Cosets and Lagrange's Theorem
3.3 The Isomorphism Theorems
3.4 Composition Series and the Holder Program
3.5 Transpositions and the Alternating Group

• 4. Group Actions

4.1 Group Actions and Permutation Representations
4.2 Groups Acting on Themselves by Left Multiplication-cayley's Theorem
4.3 Groups Acting on Themselves by Conjugation-The Class Equation
4.4 Automorphisms
4.5 The Sylow Theorems
4.6 The Simplicity of An

• 5. Direct and Semidirect Products and Abelian Groups

5.1 Direct Products
5.2 The Fundamental Theorem of Finitely Generated Abelian Groups
5.3 Table of Groups of Small Order
5.4 Recognizing Direct Products
5.5 Semidirect Products

• 6. Further Topics in Group Theory

6.1 p-groups, Nilpotent Groups, and Solvable Groups Applications in Groups of Medium Order
6.2 Application
6.3 A Word on Free Groups

• Part II - RING THEORY

• 7. Introduction to Rings

7.1 Basic Definitions and Examples
7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings
7.3 Ring Homomorphisms an Quotient Rings
7.4 Properties of Ideals
7.5 Rings of Fractions
7.6 The Chinese Remainder Theorem

• 8. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains

8.1 Euclidean Domains
8.2 Principal Ideal Domains (P.I.D.s}
8.3 Unique Factorization Domains (U.F.D.s}

• 9. Polynomial Rings

9.1 Definitions and Basic Properties
9.2 Polynomial Rings over Fields I
9.3 Polynomial Rings that are Unique Factorization Domains
9.4 Irreducibility Criteria
9.5 Polynomial Rings over Fields II
9.6 Polynomials in Several Variables over a Field and Grobner Bases

• Part III - MODULES AN D VECTOR SPACES

• 10. Introduction to Module Theory

10.1 Basic Definitions and Examples
10.2 Quotient Modules and Module Homomorphisms
10.3 Generation of Modules, Direct Sums, and Free Modules
10.4 Tensor Products of Modules
10.5 Exact Sequences-Projective, Injective, and Flat Modules

• 11. Vector Spaces

11.1 Definitions and Basic Theory
11.2 The Matrix of a Linear Transformation
11.3 Dual Vector Spaces
11.4 Determinants
11.5 Tensor Algebras. Symmetric and Exterior Algebras

• 12. Modules over Principal Ideal Domains

12.1 The Basic Theory
12.2 The Rational Canonical Form
12.3 The jordan Canonical Form

• Part IV - FIELD THEORY AND GALOIS THEORY

• 13. Field Theory

13.1 Basic Theory of Field Extensions
13.2 Algebraic Extensions
13.3 Classical Straightedge and Compass Constructions
13.4 Splitting Fields and Algebraic Closures
13.5 Separable and Inseparable Extensions
13.6 Cyclotomic Polynomials and Extensions

• 14. Galois Theory

14.1 Basic Definitions
14.2 The Fundamental Theorem of Galois Theory
14.3 Finite Fields
14.4 Composite Extensions and Simple Extensions
14.5 Cyclotomic Extensions and Abelian Extensions over Q
14.6 Galois Groups of Polynomials
14.7 Solvable and Radical Extensions: lnsolvability ofthe Quintic
14.8 Computation of Galois Groups over Q
14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups

• Part V - AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOM ETRY, AND HOMOLOGICAL ALGEBRA

• 15. Commutative Rings and Algebraic Geometry

15.1 Noetherian Rings and Affine Algebraic Sets
15.3 Integral Extensions and Hilbert's Nullstellensatz
15.4 Localization
15.5 The Prime Spectrum of a Ring

• 16. Artinian Rings, Discrete Valuation Rings, and Dedekind Domains

16.1 Artinian Rings
16.2 Discrete Valuation Rings
16.3 Dedekind Domains

• 17. Introduction to Homological Algebra and Group Cohomology

17.1 Introduction to Homological Algebra-Ext and Tor
17.2 The Cohomology of Groups
17.3 Crossed Homomorphisms and H1(G, A)
17.4 Group Extensions, Factor Sets and H2(G, A)

• Part VI - INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS

• 18. Representation Theory and Character Theory

18.1 Linear Actions and Modules over Group Rings
18.2 Wedderburn's Theorem and Some Consequences
18.3 Character Theory and the Orthogonality Relations

• 19. Examples and Applications of Character Theory

19.1 Characters of Groups of Small Order
19.2 Theorems of Burnside and Hall
19.3 Introduction to the Theory of Induced Characters

• Appendix 1: Cartesian Products and Zorn's Lemma

• Appendix II: Category Theory

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