Benjamin Fine is a professor of mathematics at Fairfield University. Anthony M. Gaglione is a professor of mathematics at the United States Naval Academy. Gerhard Rosenberger is a professor of mathematics at the University of Hamburg.

A friendly introduction to the subject-no sentence is terse, and in fact useful additional statements are sprinkled throughout an argument... Suitably chosen examples are given throughout the text to illustrate the definitions, proofs and arguments, and there are also plenty of exercises at the end of each chapter for the reinforcement of understanding. It is an excellent text for a university course. -- Peter Shiu Mathematical Gazette The utmost detailed presentation of the core material, the wealth of illustrating examples, and the many outlooks for further study make this excellent algebra primer a highly welcome, useful and valuable addition to the abundant textbook literature in the field. Zentralblatt Math

• Preface1. Abstract Algebra and Algebraic Reasoning1.1. Abstract Algebra1.2. Algebraic Structures1.3. The Algebraic Method1.4. The Standard Number Systems1.5. The Integers and Induction1.6. Exercises2. Algebraic Preliminaries2.1. Sets and Set Theory2.1.1. Set Operations2.2. Functions2.3. Equivalence Relations and Factor Sets2.4. Sizes of Sets2.5. Binary Operations2.5.1. The Algebra of Sets2.6. Algebraic Structures and Isomorphisms2.7. Groups2.8. Exercises3. Rings and the Integers3.1. Rings and the Ring of Integers3.2. Some Basic Properties of Rings and Subrings3.3. Examples of Rings3.3.1. The Modular Rings: The Integers Modulo n3.3.2. Noncommutative Rings3.3.3. Rings Without Identities3.3.4. Rings of Subsets: Boolean Rings3.3.5. Direct Sums of Rings3.3.6. Summary of Examples3.4. Ring Homomorphisms and Isomorphisms3.5. Integral Domains and Ordering3.6. Mathematical Induction and the Uniqueness of Z3.7. Exercises4. Number Theory and Unique Factorization4.1. Elementary Number Theory4.2. Divisibility and Primes4.3. Greatest Common Divisors4.4. The Fundamental Theorem of Arithmetic4.5. Congruences and Modular Arithmetic4.6. Unique Factorization Domains4.7. Exercises5. Fields: The Rationals, Reals and Complexes5.1. Fields and Division Rings5.2. Construction and Uniqueness of the Rationals5.2.1. Fields of Fractions5.3. The Real Number System5.3.1. The Completeness of R (Optional)5.3.2. Characterization of R (Optional)5.3.3. The Construction of R (Optional)5.3.4. The p-adic Numbers (Optional)5.4. The Field of Complex Numbers5.4.1. Geometric Interpretation5.4.2. Polar Form and Euler's Identity5.4.3. DeMoivre's Theorem for Powers and Roots5.5. Exercises6. Basic Group Theory6.1. Groups, Subgroups and Isomorphisms6.2. Examples of Groups6.2.1. Permutations and the Symmetric Group6.2.2. Examples of Groups: Geometric Transformation Groups6.3. Subgroups and Lagrange's Theorem6.4. Generators and Cyclic Groups6.5. Exercises7. Factor Groups and the Group Isomorphism Theorems7.1. Normal Subgroups7.2. Factor Groups7.2.1. Examples of Factor Groups7.3. The Group Isomorphism Theorems7.4. Exercises8. Direct Products and Abelian Groups8.1. Direct Products of Groups8.1.1. Direct Products of Two Groups8.1.2. Direct Products of Any Finite Number of Groups8.2. Abelian Groups8.2.1. Finite Abelian Groups8.2.2. Free Abelian Groups8.2.3. The Basis Theorem for Finitely Generated Abelian Groups8.3. Exercises9. Symmetric and Alternating Groups9.1. Symmetric Groups and Cycle Structure9.1.1. The Alternating Groups9.1.2. Conjugation in Sn9.2. The Simplicity of An9.3. Exercises10. Group Actions and Topics in Group Theory10.1. Group Actions10.2. Conjugacy Classes and the Class Equation10.3. The Sylow Theorems10.3.1. Some Applications of the Sylow Theorems10.4. Groups of Small Order10.5. Solvability and Solvable Groups10.5.1. Solvable Groups10.5.2. The Derived Series10.6. Composition Series and the Jordan-Holder Theorem10.7. Exercises11. Topics in Ring Theory11.1. Ideals in Rings11.2. Factor Rings and the Ring Isomorphism Theorem11.3. Prime and Maximal Ideals11.3.1. Prime Ideals and Integral Domains11.3.2. Maximal Ideals and Fields11.4. Principal Ideal Domains and Unique Factorization11.5. Exercises12. Polynomials and Polynomial Rings12.1. Polynomials and Polynomial Rings12.2. Polynomial Rings over a Field12.2.1. Unique Factorization of Polynomials12.2.2. Euclidean Domains12.2.3. F[x] as a Principal Ideal Domain12.2.4. Polynomial Rings over Integral Domains12.3. Zeros of Polynomials12.3.1. Real and Complex Polynomials12.3.2. The Fundamental Theorem of Algebra12.3.3. The Rational Roots Theorem12.3.4. Solvability by Radicals12.3.5. Algebraic and Transcendental Numbers12.4. Unique Factorization in Z[x]12.5. Exercises13. Algebraic Linear Algebra13.1. Linear Algebra13.1.1. Vector Analysis in R313.1.2. Matrices and Matrix Algebra13.1.3. Systems of Linear Equations13.1.4. Determinants13.2. Vector Spaces over a Field13.2.1. Euclidean n-Space13.2.2. Vector Spaces13.2.3. Subspaces13.2.4. Bases and Dimension13.2.5. Testing for Bases in Fn13.3. Dimension and Subspaces13.4. Algebras13.5. Inner Product Spaces13.5.1. Banach and Hilbert Spaces13.5.2. The Gram-Schmidt Process and Orthonormal Bases13.5.3. The Closest Vector Theorem13.5.4. Least-Squares Approximation13.6. Linear Transformations and Matrices13.6.1. Matrix of a Linear Transformation13.6.2. Linear Operators and Linear Functionals13.7. Exercises14. Fields and Field Extensions14.1. Abstract Algebra and Galois Theory14.2. Field Extensions14.3. Algebraic Field Extensions14.4. F-automorphisms, Conjugates and Algebraic Closures14.5. Adjoining Roots to Fields14.6. Splitting Fields and Algebraic Closures14.7. Automorphisms and Fixed Fields14.8. Finite Fields14.9. Transcendental Extensions14.10. Exercises15. A Survey of Galois Theory15.1. An Overview of Galois Theory15.2. Galois Extensions15.3. Automorphisms and the Galois Group15.4. The Fundamental Theorem of Galois Theory15.5. A Proof of the Fundamental Theorem of Algebra15.6. Some Applications of Galois Theory15.6.1. The Insolvability of the Quintic15.6.2. Some Ruler and Compass Constructions15.6.3. Algebraic Extensions of R15.7. ExercisesBibliographyIndex

Benjamin Fine is a professor of mathematics at Fairfield University. Anthony M. Gaglione is a professor of mathematics at the United States Naval Academy. Gerhard Rosenberger is a professor of mathematics at the University of Hamburg.

A friendly introduction to the subject-no sentence is terse, and in fact useful additional statements are sprinkled throughout an argument... Suitably chosen examples are given throughout the text to illustrate the definitions, proofs and arguments, and there are also plenty of exercises at the end of each chapter for the reinforcement of understanding. It is an excellent text for a university course. -- Peter Shiu Mathematical Gazette The utmost detailed presentation of the core material, the wealth of illustrating examples, and the many outlooks for further study make this excellent algebra primer a highly welcome, useful and valuable addition to the abundant textbook literature in the field. Zentralblatt Math

• Preface1. Abstract Algebra and Algebraic Reasoning1.1. Abstract Algebra1.2. Algebraic Structures1.3. The Algebraic Method1.4. The Standard Number Systems1.5. The Integers and Induction1.6. Exercises2. Algebraic Preliminaries2.1. Sets and Set Theory2.1.1. Set Operations2.2. Functions2.3. Equivalence Relations and Factor Sets2.4. Sizes of Sets2.5. Binary Operations2.5.1. The Algebra of Sets2.6. Algebraic Structures and Isomorphisms2.7. Groups2.8. Exercises3. Rings and the Integers3.1. Rings and the Ring of Integers3.2. Some Basic Properties of Rings and Subrings3.3. Examples of Rings3.3.1. The Modular Rings: The Integers Modulo n3.3.2. Noncommutative Rings3.3.3. Rings Without Identities3.3.4. Rings of Subsets: Boolean Rings3.3.5. Direct Sums of Rings3.3.6. Summary of Examples3.4. Ring Homomorphisms and Isomorphisms3.5. Integral Domains and Ordering3.6. Mathematical Induction and the Uniqueness of Z3.7. Exercises4. Number Theory and Unique Factorization4.1. Elementary Number Theory4.2. Divisibility and Primes4.3. Greatest Common Divisors4.4. The Fundamental Theorem of Arithmetic4.5. Congruences and Modular Arithmetic4.6. Unique Factorization Domains4.7. Exercises5. Fields: The Rationals, Reals and Complexes5.1. Fields and Division Rings5.2. Construction and Uniqueness of the Rationals5.2.1. Fields of Fractions5.3. The Real Number System5.3.1. The Completeness of R (Optional)5.3.2. Characterization of R (Optional)5.3.3. The Construction of R (Optional)5.3.4. The p-adic Numbers (Optional)5.4. The Field of Complex Numbers5.4.1. Geometric Interpretation5.4.2. Polar Form and Euler's Identity5.4.3. DeMoivre's Theorem for Powers and Roots5.5. Exercises6. Basic Group Theory6.1. Groups, Subgroups and Isomorphisms6.2. Examples of Groups6.2.1. Permutations and the Symmetric Group6.2.2. Examples of Groups: Geometric Transformation Groups6.3. Subgroups and Lagrange's Theorem6.4. Generators and Cyclic Groups6.5. Exercises7. Factor Groups and the Group Isomorphism Theorems7.1. Normal Subgroups7.2. Factor Groups7.2.1. Examples of Factor Groups7.3. The Group Isomorphism Theorems7.4. Exercises8. Direct Products and Abelian Groups8.1. Direct Products of Groups8.1.1. Direct Products of Two Groups8.1.2. Direct Products of Any Finite Number of Groups8.2. Abelian Groups8.2.1. Finite Abelian Groups8.2.2. Free Abelian Groups8.2.3. The Basis Theorem for Finitely Generated Abelian Groups8.3. Exercises9. Symmetric and Alternating Groups9.1. Symmetric Groups and Cycle Structure9.1.1. The Alternating Groups9.1.2. Conjugation in Sn9.2. The Simplicity of An9.3. Exercises10. Group Actions and Topics in Group Theory10.1. Group Actions10.2. Conjugacy Classes and the Class Equation10.3. The Sylow Theorems10.3.1. Some Applications of the Sylow Theorems10.4. Groups of Small Order10.5. Solvability and Solvable Groups10.5.1. Solvable Groups10.5.2. The Derived Series10.6. Composition Series and the Jordan-Holder Theorem10.7. Exercises11. Topics in Ring Theory11.1. Ideals in Rings11.2. Factor Rings and the Ring Isomorphism Theorem11.3. Prime and Maximal Ideals11.3.1. Prime Ideals and Integral Domains11.3.2. Maximal Ideals and Fields11.4. Principal Ideal Domains and Unique Factorization11.5. Exercises12. Polynomials and Polynomial Rings12.1. Polynomials and Polynomial Rings12.2. Polynomial Rings over a Field12.2.1. Unique Factorization of Polynomials12.2.2. Euclidean Domains12.2.3. F[x] as a Principal Ideal Domain12.2.4. Polynomial Rings over Integral Domains12.3. Zeros of Polynomials12.3.1. Real and Complex Polynomials12.3.2. The Fundamental Theorem of Algebra12.3.3. The Rational Roots Theorem12.3.4. Solvability by Radicals12.3.5. Algebraic and Transcendental Numbers12.4. Unique Factorization in Z[x]12.5. Exercises13. Algebraic Linear Algebra13.1. Linear Algebra13.1.1. Vector Analysis in R313.1.2. Matrices and Matrix Algebra13.1.3. Systems of Linear Equations13.1.4. Determinants13.2. Vector Spaces over a Field13.2.1. Euclidean n-Space13.2.2. Vector Spaces13.2.3. Subspaces13.2.4. Bases and Dimension13.2.5. Testing for Bases in Fn13.3. Dimension and Subspaces13.4. Algebras13.5. Inner Product Spaces13.5.1. Banach and Hilbert Spaces13.5.2. The Gram-Schmidt Process and Orthonormal Bases13.5.3. The Closest Vector Theorem13.5.4. Least-Squares Approximation13.6. Linear Transformations and Matrices13.6.1. Matrix of a Linear Transformation13.6.2. Linear Operators and Linear Functionals13.7. Exercises14. Fields and Field Extensions14.1. Abstract Algebra and Galois Theory14.2. Field Extensions14.3. Algebraic Field Extensions14.4. F-automorphisms, Conjugates and Algebraic Closures14.5. Adjoining Roots to Fields14.6. Splitting Fields and Algebraic Closures14.7. Automorphisms and Fixed Fields14.8. Finite Fields14.9. Transcendental Extensions14.10. Exercises15. A Survey of Galois Theory15.1. An Overview of Galois Theory15.2. Galois Extensions15.3. Automorphisms and the Galois Group15.4. The Fundamental Theorem of Galois Theory15.5. A Proof of the Fundamental Theorem of Algebra15.6. Some Applications of Galois Theory15.6.1. The Insolvability of the Quintic15.6.2. Some Ruler and Compass Constructions15.6.3. Algebraic Extensions of R15.7. ExercisesBibliographyIndex

Benjamin Fine is a professor of mathematics at Fairfield University. Anthony M. Gaglione is a professor of mathematics at the United States Naval Academy. Gerhard Rosenberger is a professor of mathematics at the University of Hamburg.

A friendly introduction to the subject-no sentence is terse, and in fact useful additional statements are sprinkled throughout an argument... Suitably chosen examples are given throughout the text to illustrate the definitions, proofs and arguments, and there are also plenty of exercises at the end of each chapter for the reinforcement of understanding. It is an excellent text for a university course. -- Peter Shiu Mathematical Gazette The utmost detailed presentation of the core material, the wealth of illustrating examples, and the many outlooks for further study make this excellent algebra primer a highly welcome, useful and valuable addition to the abundant textbook literature in the field. Zentralblatt Math

• Preface1. Abstract Algebra and Algebraic Reasoning1.1. Abstract Algebra1.2. Algebraic Structures1.3. The Algebraic Method1.4. The Standard Number Systems1.5. The Integers and Induction1.6. Exercises2. Algebraic Preliminaries2.1. Sets and Set Theory2.1.1. Set Operations2.2. Functions2.3. Equivalence Relations and Factor Sets2.4. Sizes of Sets2.5. Binary Operations2.5.1. The Algebra of Sets2.6. Algebraic Structures and Isomorphisms2.7. Groups2.8. Exercises3. Rings and the Integers3.1. Rings and the Ring of Integers3.2. Some Basic Properties of Rings and Subrings3.3. Examples of Rings3.3.1. The Modular Rings: The Integers Modulo n3.3.2. Noncommutative Rings3.3.3. Rings Without Identities3.3.4. Rings of Subsets: Boolean Rings3.3.5. Direct Sums of Rings3.3.6. Summary of Examples3.4. Ring Homomorphisms and Isomorphisms3.5. Integral Domains and Ordering3.6. Mathematical Induction and the Uniqueness of Z3.7. Exercises4. Number Theory and Unique Factorization4.1. Elementary Number Theory4.2. Divisibility and Primes4.3. Greatest Common Divisors4.4. The Fundamental Theorem of Arithmetic4.5. Congruences and Modular Arithmetic4.6. Unique Factorization Domains4.7. Exercises5. Fields: The Rationals, Reals and Complexes5.1. Fields and Division Rings5.2. Construction and Uniqueness of the Rationals5.2.1. Fields of Fractions5.3. The Real Number System5.3.1. The Completeness of R (Optional)5.3.2. Characterization of R (Optional)5.3.3. The Construction of R (Optional)5.3.4. The p-adic Numbers (Optional)5.4. The Field of Complex Numbers5.4.1. Geometric Interpretation5.4.2. Polar Form and Euler's Identity5.4.3. DeMoivre's Theorem for Powers and Roots5.5. Exercises6. Basic Group Theory6.1. Groups, Subgroups and Isomorphisms6.2. Examples of Groups6.2.1. Permutations and the Symmetric Group6.2.2. Examples of Groups: Geometric Transformation Groups6.3. Subgroups and Lagrange's Theorem6.4. Generators and Cyclic Groups6.5. Exercises7. Factor Groups and the Group Isomorphism Theorems7.1. Normal Subgroups7.2. Factor Groups7.2.1. Examples of Factor Groups7.3. The Group Isomorphism Theorems7.4. Exercises8. Direct Products and Abelian Groups8.1. Direct Products of Groups8.1.1. Direct Products of Two Groups8.1.2. Direct Products of Any Finite Number of Groups8.2. Abelian Groups8.2.1. Finite Abelian Groups8.2.2. Free Abelian Groups8.2.3. The Basis Theorem for Finitely Generated Abelian Groups8.3. Exercises9. Symmetric and Alternating Groups9.1. Symmetric Groups and Cycle Structure9.1.1. The Alternating Groups9.1.2. Conjugation in Sn9.2. The Simplicity of An9.3. Exercises10. Group Actions and Topics in Group Theory10.1. Group Actions10.2. Conjugacy Classes and the Class Equation10.3. The Sylow Theorems10.3.1. Some Applications of the Sylow Theorems10.4. Groups of Small Order10.5. Solvability and Solvable Groups10.5.1. Solvable Groups10.5.2. The Derived Series10.6. Composition Series and the Jordan-Holder Theorem10.7. Exercises11. Topics in Ring Theory11.1. Ideals in Rings11.2. Factor Rings and the Ring Isomorphism Theorem11.3. Prime and Maximal Ideals11.3.1. Prime Ideals and Integral Domains11.3.2. Maximal Ideals and Fields11.4. Principal Ideal Domains and Unique Factorization11.5. Exercises12. Polynomials and Polynomial Rings12.1. Polynomials and Polynomial Rings12.2. Polynomial Rings over a Field12.2.1. Unique Factorization of Polynomials12.2.2. Euclidean Domains12.2.3. F[x] as a Principal Ideal Domain12.2.4. Polynomial Rings over Integral Domains12.3. Zeros of Polynomials12.3.1. Real and Complex Polynomials12.3.2. The Fundamental Theorem of Algebra12.3.3. The Rational Roots Theorem12.3.4. Solvability by Radicals12.3.5. Algebraic and Transcendental Numbers12.4. Unique Factorization in Z[x]12.5. Exercises13. Algebraic Linear Algebra13.1. Linear Algebra13.1.1. Vector Analysis in R313.1.2. Matrices and Matrix Algebra13.1.3. Systems of Linear Equations13.1.4. Determinants13.2. Vector Spaces over a Field13.2.1. Euclidean n-Space13.2.2. Vector Spaces13.2.3. Subspaces13.2.4. Bases and Dimension13.2.5. Testing for Bases in Fn13.3. Dimension and Subspaces13.4. Algebras13.5. Inner Product Spaces13.5.1. Banach and Hilbert Spaces13.5.2. The Gram-Schmidt Process and Orthonormal Bases13.5.3. The Closest Vector Theorem13.5.4. Least-Squares Approximation13.6. Linear Transformations and Matrices13.6.1. Matrix of a Linear Transformation13.6.2. Linear Operators and Linear Functionals13.7. Exercises14. Fields and Field Extensions14.1. Abstract Algebra and Galois Theory14.2. Field Extensions14.3. Algebraic Field Extensions14.4. F-automorphisms, Conjugates and Algebraic Closures14.5. Adjoining Roots to Fields14.6. Splitting Fields and Algebraic Closures14.7. Automorphisms and Fixed Fields14.8. Finite Fields14.9. Transcendental Extensions14.10. Exercises15. A Survey of Galois Theory15.1. An Overview of Galois Theory15.2. Galois Extensions15.3. Automorphisms and the Galois Group15.4. The Fundamental Theorem of Galois Theory15.5. A Proof of the Fundamental Theorem of Algebra15.6. Some Applications of Galois Theory15.6.1. The Insolvability of the Quintic15.6.2. Some Ruler and Compass Constructions15.6.3. Algebraic Extensions of R15.7. ExercisesBibliographyIndex