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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