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A First Course in Abstract Algebra: Rings, Groups, and Fields (2nd Edition) by Marlow Anderson, Todd Feil



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A First Course in Abstract Algebra: Rings, Groups, and Fields (2nd Edition) written by Marlow Anderson, Colorado College, Colorado Springs, USA and Todd Feil, Denison University, Granville, Ohio, USA. Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there is more natural-and ultimately more effective. Authors Anderson and Feil developed A First Course in Abstract Algebra: Rings, Groups and Fields based upon that conviction. The text begins with ring theory, building upon students' familiarity with integers and polynomials. Later, when students have become more experienced, it introduces groups. The last 'of the book develops Galois Theory with the goal of showing the impossibility of solving the quintic with radicals. Each section of the book ends with a "Section in a Nutshell" synopsis of important definitions and theorems. Each chapter includes "Quick Exercises" that reinforce the topic addressed and are designed to be worked as the text is read. Problem sets at the end of each chapter begin with "Warm-Up Exercises" that test fundamental comprehension, followed by regular exercises, both computational and "supply the proof" problems. A Hints and Answers section is provided at the end of the book. As stated in the title, this book is designed for a first course--either one or two semesters in abstract algebra. It requires only a typical calculus sequence as a prerequisite and does not assume any familiarity with linear algebra or complex numbers.

A First Course in Abstract Algebra: Rings, Groups, and Fields (2nd Edition) written by Marlow Anderson and Todd Feil cover the following topics.

  • I Numbers, Polynomials, and Factoring

  • 1. The Natural Numbers

  • 2. The Integers

  • 3. Modular Arithmetic

  • 4. Polynomials with Rational Coecients

  • 5. Factorization of Polynomials

  • II Rings, Domains, and Fields

  • 6. Rings

  • 7. Subrings and Unity

  • 8. Integral Domains and Fields

  • 9. Polynomials over a Field

  • III Unique Factorization

  • 10. Associates and Irreducebles

  • 11. Factorization and Ideal

  • 12. Principal Ideal Domain

  • 13. Prime and Unique Factorization

  • 14. Polynomials with Integer Coecients

  • 15. Euclidean Domains

  • IV Ring Homomorphisms and Ideals

  • 16. Ring Homomorphisms

  • 17. The Kernel

  • 18. Rings of Cosets

  • 19. The Isomorphism Theorem for Rings

  • 20. Maximal and Prime Ideals

  • 21. The Chinese Remainder Theorem

  • V Groups

  • 22. Symmetries of Geometric Figures

  • 23. Permutations

  • 24. Abstract Groups

  • 25. Subgroups

  • 26. Cyclic Groups

  • VI Group Homomorphisms

  • 27. Group Homomorphisms

  • 28. Structure and Representation

  • 31. Cosets and Lagrange's Theorem

  • 32. Groups of Cosets

  • 33. The Isomorphism Theorem for Groups

  • VI Topics from Group Theory

  • 34. The Alternating Groups

  • 35. Fundamental Theorem of Abelian Groups

  • 36. Solvable Groups

  • VII Constructibility Problems

  • 37. Constructions with Compass and Straightedge

  • 38. Constructibility and Quadratic Field Extensions

  • 39. The Impossibility of Certain Constructions

  • VIII Vector Spaces and Field Extensions

  • 40. Vector Spaces I

  • 41. Vector Spaces II

  • 42. Field Extensions and Kronecker's Theorem

  • 43. Algebraic Field Extensions

  • 44. Finite Extensions and Constructibility Revisited

  • X Galois Theory

  • 45. The Splitting Field

  • 46. Finite Fields

  • 47. Galois Groups

  • 48. The Fundamental Theorem of Galois Theory

  • 49. Solving Polynomials by Radicals

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