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A First Course in Abstract Algebra: Rings, Groups, and Fields 3/e

作者：Marlow Anderson, Todd Feil
原價：NT$ 1,400

ISBN：9781482245523
版次：3
年份：2014
出版商：Taylor & Francis
頁數/規格：536頁/精裝單色
參考網頁：A First Course in Abstract Algebra: Rings, Groups, and Fields 3/e

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內容介紹本書特色目錄

Description
Like its popular predecessors, A First Course in Abstract Algebra: Rings, Groups, and Fields, Third Edition develops ring theory first by drawing on students’ familiarity with integers and polynomials. This unique approach motivates students in the study of abstract algebra and helps them understand the power of abstraction. The authors introduce groups later on using examples of symmetries of figures in the plane and space as well as permutations.

The text includes straightforward exercises within each chapter for students to quickly verify facts, warm-up exercises following the chapter that test fundamental comprehension, and regular exercises concluding the chapter that consist of computational and supply-the-proof problems. Historical remarks discuss the history of algebra to underscore certain pedagogical points. Each section also provides a synopsis that presents important definitions and theorems, allowing students to verify the major topics from the section.

Features

Offers options for using the text in a one- or two-semester undergraduate course

Uses integers and polynomials as the motivating examples for studying rings first

Introduces the important proof technique of induction

Presents many examples that illustrate the power of abstract algebra

Covers the fundamental isomorphism theorem and Sylow theorems

Demonstrates the impossibility of solving the quintic equation with radicals

Contains a variety of exercises, including computational problems, with some hints and solutions at the back of the book

New To This Edition

Makes it easier to teach unique factorization as an optional topic

Reorganizes the core material on rings, integral domains, and fields

Includes a more detailed treatment of permutations

Introduces more topics in group theory, including new chapters on Sylow theorems

Provides many new exercises on Galois theory

Table of Contents
I. Numbers, Polynomials, and Factoring
1. The Natural Numbers
2. The Integers
3. Modular Arithmetic
4. Polynomials with Rational Coefficients
5. Factorization of Polynomials

II. Rings, Domains, and Fields
6. Rings
7. Subrings and Unity
8. Integral Domains and Fields
9. Ideals
10. Polynomials over a Field

III. Ring Homomorphisms and Ideals
11. Ring Homomorphisms
12. The Kernel
13. Rings of Cosets
14. The Isomorphism Theorem for Rings
15. Maximal and Prime Ideals
16. The Chinese Remainder Theorem

IV. Groups
17. Symmetries of Geometric Figures
18. Permutations
19. Abstract Groups
20. Subgroups
21. Cyclic Groups

V. Group Homomorphisms
22. Group Homomorphisms
23. Structure and Representation
24. Cosets and Lagrange's Theorem
25. Groups of Cosets
26. The Isomorphism Theorem for Groups

VI. Topics from Group Theory
27. The Alternating Groups
28. Sylow Theory: The Preliminaries
29. Sylow Theory: The Theorems
30. Solvable Groups

VII. Unique Factorization
31. Quadratic Extensions of the Integers
32. Factorization
33. Unique Factorization
34. Polynomials with Integer Coefficients
35. Euclidean Domains

VIII. Constructibility Problems
36. Constructions with Compass and Straightedge
37. Constructibility and Quadratic Field Extensions
38. The Impossibility of Certain Constructions

IX. Vector Spaces and Field Extensions
39. Vector Spaces I
40. Vector Spaces II
41. Field Extensions and Kronecker's Theorem
42. Algebraic Field Extensions
43. Finite Extensions and Constructibility Revisited

X. Galois Theory
44. The Splitting Field
45. Finite Fields
46. Galois Groups
47. The Fundamental Theorem of Galois Theory
48. Solving Polynomials by Radicals

Section X in a Nutshell
Hints and Solutions
Guide to Notation
Index

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