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.
Shahriar Shahriari, Pomona College, Claremont, CA
Contents Preface
Part 1. (Mostly Finite) Group Theory Chapter 1. Four Basic Examples 1.1. Symmetries of a Square 1.2. 1-1 and Onto Functions 1.3. Integers \bmodn and Elementary Properties of Integers 1.4. Invertible Matrices 1.5. More Problems and Projects
Chapter 2. Groups: The Basics 2.1. Definitions and Examples 2.2. Cancellation Properties 2.3. Cyclic Groups and the Order of an Element 2.4. Isomorphisms 2.5. Direct Products (New Groups from Old Groups) 2.6. Subgroups 2.7. More Problems and Projects
Chapter 3. The Alternating Groups 3.1. Permutations, Cycles, and Transpositions 3.2. Even and Odd Permutations and A_{n} 3.3. More Problems and Projects
Chapter 4. Group Actions 4.1. Definition and Examples 4.2. The Cayley Graph of a Group Action* 4.3. Stabilizers 4.4. Orbits 4.5. More Problems and Projects
Chapter 5. A Subgroup Acts on the Group: Cosets and Lagrange’s Theorem 5.1. Translation Action and Cosets 5.2. Lagrange’s Theorem 5.3. Application to Number Theory^{⋆} 5.4. More Problems and Projects
Chapter 6. A Group Acts on Itself: Counting and the Conjugation Action 6.1. The Fundamental Counting Principle 6.2. The Conjugation Action 6.3. The Class Equation and Groups of Order p2 6.4. More Problems and Projects
Chapter 7. Acting on Subsets, Cosets, and Subgroups: The Sylow Theorems 7.1. Binomial Coefficients \bmod𝑝 7.2. The Sylow E(xistence) Theorem 7.3. The Number and Conjugacy of Sylow Subgroups^{⋆}
Chapter 8. Counting the Number of Orbits^{⋆} 8.1. The Cauchy–Frobenius Counting Lemma 8.2. Combinatorial Applications of the Counting Lemma 8.3. More Problems and Projects
Chapter 9. The Lattice of Subgroups^{⋆} 9.1. Partially Ordered Sets, Hasse Diagrams, and Lattices 9.2. Edge Lengths and Partial Lattice Diagrams 9.3. More Problems and Projects
Chapter 10. Acting on Its Subgroups: Normal Subgroups and Quotient Groups 10.1. Normal Subgroups 10.2. The Normalizer 10.3. Quotient Groups 10.4. More Problems and Projects
Chapter 11. Group Homomorphisms 11.1. Definitions, Examples, and Elementary Properties 11.2. The Kernel and the Image 11.3. Homomorphisms, Normal Subgroups, and Quotient Groups 11.4. Actions and Homomorphisms 11.5. The Homomorphism Theorems 11.6. Automorphisms and Inner-automorphisms^{⋆} 11.7. More Problems and Projects
Chapter 12. Using Sylow Theorems to Analyze Finite Groups* 12.1. 𝑝-groups 12.2. Acting on Cosets and Existence of Normal Subgroups 12.3. Applying the Sylow Theorems 12.4. 𝐴₅ Is the Only Simple Group of Order 60
Chapter 13. Direct and Semidirect Products^{⋆} 13.1. Direct Products of Groups 13.2. Fundamental Theorem of Finite Abelian Groups 13.3. Semidirect Products 13.4. Groups of Very Small Order
Chapter 14. Solvable and Nilpotent Groups^{⋆} 14.1. Solvable Groups 14.2. Nilpotent Groups 14.3. The Jordan–Hölder Theorem
Part 2. (Mostly Commutative) Ring Theory Chapter 15. Rings 15.1. Diophantine Equations and Rings 15.2. Rings, Integral Domains, Division Rings, and Fields 15.3. Finite Integral Domains
Chapter 16. Homomorphisms, Ideals, and Quotient Rings 16.1. Subrings, Homomorphisms, and Ideals 16.2. Quotient Rings and Homomorphism Theorems 16.3. Characteristic of Rings with Identity, Integral Domains, and Fields 16.4. Manipulating Ideals^{⋆}
Chapter 17. Field of Fractions and Localization 17.1. Field of Fractions and Localization of an Integral Domain 17.2. Localization of Commutative Rings with Identity^{⋆}
Chapter 18. Factorization, EDs, PIDs, and UFDs 18.1. Factorization in Commutative Rings 18.2. Ascending Chain Condition and Noetherian Rings 18.3. A PID is a UFD 18.4. Euclidean Domains 18.5. The Greatest Common Divisor^{⋆} 18.6. More Problems and Projects
Chapter 19. Polynomial Rings 19.1. Polynomials 19.2. 𝐾 a field ⇒ 𝐾[𝑥] an ED 19.3. Roots of Polynomials and Construction of Finite Fields 19.4. 𝑅 UFD ⇒ 𝑅[𝑥] UFD and Gauss’s Lemma 19.5. Irreducibility Criteria 19.6. Hilbert Basis Theorem^{⋆} 19.7. More Problems and Projects
Chapter 20. Gaussian Integers and (a little) Number Theory^{⋆} 20.1. Gaussian Integers 20.2. Unique Factorization and Diophantine Equations
Part 3. Fields and Galois Theory Chapter 21. Introducing Field Theory and Galois Theory 21.1. The Classical Problems of Field Theory 21.2. Roots of Equations, Fields, and Groups—An Example 21.3. A Quick Review of Ring Theory
Chapter 22. Field Extensions 22.1. Simple and Algebraic Extensions 22.2. A Quick Review of Vector Spaces 22.3. The Degree of an Extension
Chapter 23. Straightedge and Compass Constructions 23.1. The Field of Constructible Numbers 23.2. Characterizing Constructible Numbers
Chapter 24. Splitting Fields and Galois Groups 24.1. Roots of Polynomials, Field Extensions, and 𝐹-isomorphisms 24.2. Splitting Fields 24.3. Galois Groups and Their Actions on Roots
Chapter 25. Galois, Normal, and Separable Extensions 25.1. Subgroups of the Galois Group and Intermediate Fields 25.2. Galois, Normal, and Separable Extensions 25.3. More on Normal Extensions 25.4. More on Separable Extensions 25.5. Simple Extensions 25.6. More Problems and Projects
Chapter 26. Fundamental Theorem of Galois Theory 26.1. Galois Groups and Fixed Fields 26.2. Fundamental Theorem of Galois Theory 26.3. Examples of Galois Groups
Chapter 27. Finite Fields and Cyclotomic Extensions 27.1. Finite Fields 27.2. Cyclotomic Extensions 27.3. The Polynomial 𝑥ⁿ-𝑎 27.4. More Problems and Projects
Chapter 28. Radical Extensions, Solvable Groups, and the Quintic 28.1. Solvability by Radicals 28.2. A Solvable Polynomial Has a Solvable Galois Group 28.3. A Solvable Galois Group Corresponds to a Solvable Polynomial^{⋆} 28.4. More Problems and Projects
Appendix A. Hints for Selected Problems Appendix B. Short Answers for Selected Problems Appendix C. Complete Solutions for Selected (Odd-Numbered) Problems Bibliography Index