Algebra in Action

این متن بر اساس دوره‌های محبوب نویسنده در دانشگاه پومونا نوشته شده است و یک مقدمه خوانا، دوستانه برای دانشجویان و کمی پیشرفته به جبر انتزاعی ارائه می‌دهد.

هدف این کتاب دانشجویان سال دوم یا سوم کارشناسی است که برای اولین بار با این مباحث روبرو می‌شوند. علاوه بر تعاریف و قضایای معمول، بحث‌های گسترده‌ای برای کمک به دانشجویان در ساختن شهود و یادگیری نحوه تفکر درباره مفاهیم انتزاعی وجود دارد. این کتاب بیش از 1300 تمرین و پروژه کوچک با درجات مختلف سختی دارد و برای تسهیل یادگیری فعال و مطالعه خودآموز، نکات و پاسخ‌های کوتاهی برای بسیاری از مسائل ارائه شده است.

همچنین برای بیش از 100 مسئله، راه‌حل‌های کامل آورده شده است تا متن را تکمیل کند و نوشتن راه‌حل‌ها را مدل‌سازی نماید.

نمودارهای شبکه‌ای در سراسر کتاب برای نمایش بصری نتایج و تکنیک‌های اثبات استفاده می‌شوند. کتاب به بررسی گروه‌ها، حلقه‌ها و میدان‌ها می‌پردازد. در نظریه گروه‌ها، عمل‌های گروهی به عنوان تم متحدکننده معرفی می‌شوند و در اوایل کتاب معرفی می‌شوند. نظریه حلقه‌ها از آنچه برای حل معادلات دیوفانتینی لازم است انگیزه می‌گیرد و در نظریه میدان‌ها، نظریه گالوآ و حل‌پذیری چندجمله‌ای‌ها در کانون توجه قرار دارند.

در هر زمینه، کتاب به اندازه کافی عمیق می‌رود تا قدرت تفکر انتزاعی را نشان دهد و خواننده را متقاعد کند که این موضوع پر از نتایج غیرمنتظره است.

This text-based on the author’s popular courses at Pomona College-provides a readable, student-friendly, and somewhat sophisticated introduction to abstract algebra. It is aimed at sophomore or junior undergraduates who are seeing the material for the first time. In addition to the usual definitions and theorems, there is ample discussion to help students build intuition and learn how to think about the abstract concepts. The book has over 1300 exercises and mini-projects of varying degrees of difficulty, and, to facilitate active learning and self-study, hints and short answers for many of the problems are provided. There are full solutions to over 100 problems in order to augment the text and to model the writing of solutions. Lattice diagrams are used throughout to visually demonstrate results and proof techniques. The book covers groups, rings, and fields. In group theory, group actions are the unifying theme and are introduced early. Ring theory is motivated by what is needed for solving Diophantine equations, and, in field theory, Galois theory and the solvability of polynomials take center stage. In each area, the text goes deep enough to demonstrate the power of abstract thinking and to convince the reader that the subject is full of unexpected results.

Table of Contents

Part 1: (Mostly Finite) Group Theory

1. Four Basic Examples

2. Groups: The Basics

3. The Alternating Groups

4. Group Actions

5. A Subgroup Acts on the Group: Cosets and Lagrange’s Theorem

6. A Group Acts on Itself: Counting and the Conjugation Action

7. Acting on Subsets, Cosets, and Subgroups: The Sylow Theorems

8. Counting the Number of Orbits

9. The Lattice of Subgroups

10. Acting on Its Subgroups: Normal Subgroups and Quotient Groups

11. Group Homomorphisms

12. Using Sylow Theorems to Analyze Finite Groups

13. Direct and Semidirect Products

14. Solvable and Nilpotent Groups

Part 2: (Mostly Commutative) Ring Theory

15. Rings

16. Homomorphisms, Ideals, and Quotient Rings

17. Field of Fractions and Localization

18. Factorization, EDs, PIDs, and UFDs

19. Polynomial Rings

20. Gaussian Integers and (a little) Number Theory

Part 3: Fields and Galois Theory

21. Introducing Field Theory and Galois Theory

22. Field Extensions

23. Straightedge and Compass Constructions

24. Splitting Fields and Galois Groups

25. Galois, Normal, and Separable Extensions

26. Fundamental Theorem of Galois Theory

27. Finite Fields and Cyclotomic Extensions

28. Radical Extensions, Solvable Groups, and the Quintic

Appendix A: Hints for Selected Problems

Appendix B: Short Answers for Selected Problems

Appendix C: Complete Solutions for Selected (Odd-Numbered) Problems

Review

Written with great care and clarity, Shahriari’s “Algebra in Action” provides an excellent introduction to abstract algebra. I have used the book twice to teach abstract algebra class at Reed College, and it’s a perfect fit. The book is sophisticated yet readable, and packed with examples and exercises. Group actions appear early on, serving to motivate and unify many of the important concepts in group theory. The book also includes plenty of material on rings and fields, including the basics of Galois theory. –Jamie Pommersheim, Reed College

Shahriar Shahriari has written an exquisite text that will become, and deserves to be, widely used for introducing generations of students to abstract algebra.The presentation is engaging, modern, and sufficiently detailed, making the book ideal for self-study…”Algebra in Action” is a gem and, no doubt, it is the work of a master teacher whose passion and respect for the subject is apparent everywhere in the book. I highly recommend it to students and professors alike! –Ehssan Khanmohammadi

The structure of the text “Algebra in Action” lets students see what groups really do right from the very beginning. In the very first chapter, the author introduces a rich selection of examples, the dihedral groups, the symmetric group, the integers modulo n, and matrix groups, that students can see ‘in action’ before the presentation of the formal definitions of groups and group actions in chapter 2 where the theoretical foundations are introduced. Students return to these examples again and again as the formal theory unfolds, seeing how the theory lets them study all groups at once…It is one of the few texts at the undergraduate level that supports the incorporation of group actions at an early stage in the course. –Jessica Sidman, Mount Holyoke College

About the Author

Shahriar Shahriari, Pomona College, Claremont, CA.