A Friendly Introduction to Abstract Algebra

Description

A Friendly Introduction to Abstract Algebra offers a new approach to laying a foundation for abstract mathematics. Prior experience with proofs is not assumed, and the book takes time to build proof-writing skills in ways that will serve students through a lifetime of learning and creating mathematics. The author's pedagogical philosophy is that when students abstract from a wide range of examples, they are better equipped to conjecture, formalize, and prove new ideas in abstract algebra. Thus, students thoroughly explore all concepts through illuminating examples before formal definitions are introduced. The instruction in proof writing is similarly grounded in student exploration and experience. Throughout the book, the author carefully explains where the ideas in a given proof come from, along with hints and tips on how students can derive those proofs on their own. Readers of this text are not just consumers of mathematical knowledge. Rather, they are learning mathematics by creating mathematics. The author's gentle, helpful writing voice makes this text a particularly appealing choice for instructors and students alike. The book's website has companion materials that support the active-learning approaches in the book, including in-class modules designed to facilitate student exploration. Cover Title page Copyright Contents Preface For the student For the instructor Note about rings Road map Acknowledgments Unit I: Preliminaries Chapter 1. Introduction to Proofs 1.1. Proving an implication 1.2. Proof by cases 1.3. Contrapositive 1.4. Proof by contradiction 1.5. If and only if 1.6. Counterexample Exercises Chapter 2. Sets and Subsets 2.1. What is a set? 2.2. Set of integers and its subsets 2.3. Closure 2.4. Showing set equality Exercises Chapter 3. Divisors 3.1. Divisor 3.2. GCD theorem 3.3. Proofs involving the GCD theorem Exercises Unit II: Examples of Groups Chapter 4. Modular Arithmetic 4.1. Number system Z ₇ 4.2. Equality in Z ₇ 4.3. Multiplicative inverses Exercises Chapter 5. Symmetries 5.1. Symmetries of a square 5.2. Group properties of 𝐷₄ 5.3. Centralizer Exercises Chapter 6. Permutations 6.1. Permutations of the set {1,2,3} 6.2. Group properties of 𝑆_{𝑛} 6.3. Computations in 𝑆_{𝑛} 6.4. Associative law in 𝑆_{𝑛} (and in 𝐷_{𝑛}) Exercises Chapter 7. Matrices 7.1. Matrix arithmetic 7.2. Matrix group 𝑀(Z ₁₀) 7.3. Multiplicative inverses 7.4. Determinant Exercises Unit III: Introduction to Groups Chapter 8. Introduction to Groups 8.1. Definition of a “group” 8.2. Essential properties of a group 8.3. Proving that a group is commutative 8.4. Non-associative operations 8.5. Direct product Exercises Chapter 9. Groups of Small Size 9.1. Smallest group 9.2. Groups with two elements 9.3. Groups with three elements 9.4. Sudoku property 9.5. Groups with four elements Exercises Chapter 10. Matrix Groups 10.1. Groups Z ₁₀ and 𝑈₁₀ 10.2. Groups 𝑀(Z ₁₀) and 𝐺(Z ₁₀) 10.3. Group 𝑆(Z ₁₀) Exercises Chapter 11. Subgroups 11.1. Examples of subgroups 11.2. Subgroup proofs 11.3. Center and centralizer revisited Exercises Chapter 12. Order of an Element 12.1. Motivating example 12.2. When does 𝑔^{𝑘}=𝜖? 12.3. Conjugates 12.4. Order in an additive group 12.5. Elements with infinite order Exercises Chapter 13. Cyclic Groups, Part I 13.1. Generators of the additive group Z ₁₂ 13.2. Generators of the multiplicative group 𝑈₁₃ 13.3. Matching Z ₁₂ and 𝑈₁₃ 13.4. Taking positive and negative powers of 𝑔 13.5. When the group operation is addition Exercises Chapter 14. Cyclic Groups, Part II 14.1. Why negative powers are needed 14.2. Additive groups revisited 14.3. ⟨3⟩ behaves “just like” Z 14.4. Subgroups of cyclic groups Exercises Unit IV: Group Homomorphisms Chapter 15. Functions 15.1. Domain and codomain 15.2. One-to-one function 15.3. Onto function 15.4. When domain and codomain have the same size Exercises Chapter 16. Isomorphisms 16.1. Groups Z ₁₂ and ⟨𝑔⟩: Elements match up 16.2. Groups Z ₁₂ and ⟨𝑔⟩: Operations match up 16.3. Elements with infinite order revisited 16.4. Inverse isomorphisms Exercises Chapter 17. Homomorphisms, Part I 17.1. Group homomorphism 17.2. Properties of homomorphisms 17.3. Order of an element Exercises Chapter 18. Homomorphisms, Part II 18.1. Kernel of a homomorphism 18.2. Image of a homomorphism 18.3. Partitioning the domain 18.4. Finding homomorphisms Exercises Unit V: Quotient Groups Chapter 19. Introduction to Cosets 19.1. Multiplicative group example 19.2. Additive group example 19.3. Right cosets 19.4. Properties of cosets 19.5. When are cosets equal? Exercises Chapter 20. Lagrange’s Theorem 20.1. Motivating Lagrange’s theorem 20.2. Proving Lagrange’s theorem 20.3. Applications of Lagrange’s theorem Exercises Chapter 21. Multiplying/Adding Cosets 21.1. Turning a set of cosets into a group 21.2. Coset multiplication shortcut 21.3. Cosets of 𝐻=5Z in Z revisited Exercises Chapter 22. Quotient Group Examples 22.1. Quotient group 𝑈₁₃/𝐻 revisited 22.2. Quotient group 𝑈₃₇/𝐻 22.3. Quotient group 𝐺/𝐻 (generalization) Exercises Chapter 23. Quotient Group Proofs 23.1. Sample quotient group proofs 23.2. Collapsing 𝐺 into 𝐺/𝐻 Exercises Chapter 24. Normal Subgroups 24.1. How does the shortcut fail and work? 24.2. Normal subgroups: What and why 24.3. Examples of normal subgroups 24.4. Normal subgroup test Exercises Chapter 25. First Isomorphism Theorem 25.1. Familiar homomorphism 25.2. Another homomorphism 25.3. First Isomorphism Theorem 25.4. Finding and building homomorphisms Exercises Unit VI: Introduction to Rings Chapter 26. Introduction to Rings 26.1. Examples and definition 26.2. Fundamental properties 26.3. Units and zero divisors 26.4. Subrings 26.5. Group of units Exercises Chapter 27. Integral Domains and Fields 27.1. Integral domains 27.2. Fields 27.3. Idempotent elements Exercises Chapter 28. Polynomial Rings, Part I 28.1. Examples and definition 28.2. Degree of a polynomial 28.3. Units and zero divisors Exercises Chapter 29. Polynomial Rings, Part II 29.1. Division algorithm in 𝐹[𝑥] 29.2. Factor theorem 29.3. Nilpotent elements Big picture stuff Exercises Chapter 30. Factoring Polynomials 30.1. Examples and definition 30.2. Factorable or unfactorable? Big picture stuff Exercises Unit VII: Quotient Rings Chapter 31. Ring Homomorphisms 31.1. Evaluation map 31.2. Properties of ring homomorphisms 31.3. Kernel and image 31.4. Examples and definition of an ideal 31.5. Ideals in Z and in 𝐹[𝑥] Big picture stuff Exercises Chapter 32. Introduction to Quotient Rings 32.1. From a quotient group to a quotient ring 32.2. Role of an ideal in a quotient ring 32.3. Quotient ring Z ₃[𝑥]/⟨𝑥²⟩ 32.4. First Isomorphism Theorem for rings Big picture stuff Exercises Chapter 33. Quotient Ring Z ₇[𝑥]/⟨𝑥²-1⟩ 33.1. Division algorithm revisited 33.2. Another way to reduce in Z ₇[𝑥]/⟨𝑥²-1⟩ 33.3. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is not a field 33.4. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field Big picture stuff Exercises Chapter 34. Quotient Ring R [𝑥]/⟨𝑥²+1⟩ 34.1. Reducing elements in R [𝑥]/⟨𝑥²+1⟩ 34.2. Field of complex numbers 34.3. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field revisited Exercises Chapter 35. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ Is/Isn’t a Field, Part I 35.1. Translate from 𝐹[𝑥] to Z 35.2. Translate (back) from Z to 𝐹[𝑥] 35.3. Proof of Theorem 35.1(b) Big picture stuff Exercises Chapter 36. Maximal Ideals 36.1. Examples and definition 36.2. Maximality of ⟨𝑔(𝑥)⟩ Big picture stuff Exercises Chapter 37. 𝐹[𝑥]/⟨𝑔(𝑥)⟩ Is/Isn’t a Field, Part II 37.1. Maximal ideals and quotient rings 37.2. Putting it all together 37.3. Oh wait, but there’s more! 37.4. Prime ideals Exercises Appendix A. Proof of the GCD Theorem Appendix B. Composition Table for 𝐷₄ Appendix C. Symbols and Notations Appendix D. Essential Theorems Index of Terms Back Cover