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