Book Outline#

Here is a recommended 2-level outline for a logically thorough book on abstract algebra, based on the provided research.

Part 0: Preliminaries#

• Chapter P1: Why Abstract Algebra?
  • The Power of Abstraction
  • Historical Impetus: Solving Ancient Puzzles
  • Modern Relevance and Applications
  • The Intellectual Appeal of Structure
• Chapter P2: A Glimpse into History
  • The Quest for Solutions: Polynomial Equations and Group Theory
  • The Evolution of Core Structures: Rings, Fields, and Geometry
  • The Age of Unification and Axiomatization
  • Key Milestones Timeline
• Chapter P3: Essential Mathematical Toolkit
  • Set Theory: The Basic Language
  • Functions (Mappings): Relating Sets
  • Relations: Structuring Sets Internally
  • Properties of Integers: The Archetypal Algebraic System
  • Basic Proof Techniques
• Chapter P4: The Spirit of Axiomatics
  • The Historical Journey of the Axiomatic Method
  • Purpose and Power of Axioms in Mathematics
  • Defining Structures: Interplay Between Concrete and Abstract
  • A Comparative Overview of Axioms

Part I: Group Theory: The Archetypal Algebraic Structure#

• Chapter 1: Introduction to Groups
  • Binary Operations
  • Definition and Examples of Groups
  • Elementary Properties of Groups
  • Historical Note: The Genesis of Group Theory
• Chapter 2: Subgroups
  • Definition and Examples
  • Subgroup Criteria (Subgroup Tests)
  • Cyclic Groups and Generators
  • The Lattice of Subgroups
• Chapter 3: Permutation Groups
  • Symmetric Groups ($S_n$)
  • Cycle Notation, Transpositions, and Parity
  • Cayley's Theorem
  • Historical Note: Permutations as the First Groups
• Chapter 4: Cosets and Lagrange's Theorem
  • Cosets
  • Lagrange's Theorem and its Consequences
• Chapter 5: Normal Subgroups and Quotient Groups
  • Normal Subgroups: Definition and Properties
  • Quotient (Factor) Groups
  • Historical Note: Galois and Normal Subgroups
• Chapter 6: Homomorphisms and Isomorphisms
  • Group Homomorphisms: Definitions and Examples
  • Kernels and Images
  • The Isomorphism Theorems
  • Automorphisms
• Chapter 7: Direct Products and Finitely Generated Abelian Groups
  • External and Internal Direct Products
  • The Fundamental Theorem of Finitely Generated Abelian Groups
• Chapter 8: Group Actions
  • Definition and Examples
  • Orbits and Stabilizers (The Orbit-Stabilizer Theorem)
  • The Class Equation
  • Sylow's Theorems

Part II: Ring Theory: Generalizing Arithmetic Structures#

• Chapter 9: Introduction to Rings
  • Definition and Examples of Rings
  • Basic Properties of Rings
  • Types of Rings (Commutative, Unity, Zero Divisors)
  • Integral Domains, Division Rings, and Fields
  • Historical Note: From Number Theory to Abstract Rings
• Chapter 10: Ideals and Quotient Rings
  • Ideals: The Subobjects of Rings
  • Quotient Rings
  • Prime and Maximal Ideals
• Chapter 11: Ring Homomorphisms and Isomorphisms
  • Ring Homomorphisms, Kernels, and Images
  • The Isomorphism Theorems for Rings
• Chapter 12: Factorization in Commutative Rings
  • Divisibility, Associates, and Irreducible Elements
  • Unique Factorization Domains (UFDs)
  • Principal Ideal Domains (PIDs)
  • Euclidean Domains
  • Relationships between ED, PID, and UFD

Part III: Field Theory and Galois Theory#

• Chapter 13: Introduction to Fields
  • Field Axioms Revisited
  • Field Extensions
  • Algebraic and Transcendental Elements
  • Minimal Polynomials
• Chapter 14: Geometric Constructions
  • Constructible Numbers
  • Impossibility of Classical Problems (Squaring the Circle, etc.)
• Chapter 15: Finite Fields
  • Existence and Uniqueness
  • Structure of Finite Fields
  • Subfields of a Finite Field
• Chapter 16: Galois Theory
  • The Galois Group of a Polynomial
  • The Fundamental Theorem of Galois Theory
  • Solvability of Polynomials by Radicals

Part IV: Advanced Topics (Optional)#

• Chapter 17: Introduction to Module Theory
  • Modules: Vector Spaces over Rings
  • Submodules, Quotient Modules, and Homomorphisms
  • Modules over a Principal Ideal Domain
• Chapter 18: Further Topics in Group Theory
  • The Jordan-Hölder Theorem
  • Solvable and Nilpotent Groups
  • Introduction to Group Representations
• Chapter 19: An Introduction to Commutative Algebra
  • Noetherian Rings Revisited
  • Hilbert's Basis Theorem
  • Introduction to Algebraic Geometry