Abstract Algebra - Table of Contents

📚 Overview

Abstract Algebra is the study of algebraic structures and their properties. This comprehensive course is organized into three main parts:

1. Group Theory - The mathematics of symmetry
2. Ring and Field Theory - Beyond a single operation
3. Galois Theory - The symmetry of equations

📖 Main Resources

• README - Comprehensive guide and learning path
• Key Definitions - Complete glossary of all key terms

🗂️ Part I: Group Theory - The Mathematics of Symmetry

0 - Basic Structures in Algebra

• Basic Structures - Sets of numbers, vector spaces, algebraic structures

1 - Groups and Subgroups

• Groups and Subgroups - Group axioms, subgroups, dihedral groups, permutation groups
• Group Axioms - The four group axioms and their motivation
• Subgroups - Definition and properties of subgroups
• Dihedral Groups - Symmetry groups of regular polygons
• Symmetric Groups - Permutation groups and Cayley's theorem

2 - Cosets and Lagrange's Theorem

• Cosets and Lagrange's Theorem - Cosets, Lagrange's theorem, normal subgroups, quotient groups
• Cosets - Definition and properties of cosets
• Lagrange's Theorem - The fundamental theorem of finite group theory
• Normal Subgroups - Definition and properties of normal subgroups
• Quotient Groups - Construction and properties of quotient groups

3 - Isomorphism Theorems

• Isomorphism Theorems - Three isomorphism theorems and their applications
• Group Homomorphisms - Definition and properties of group homomorphisms
• First Isomorphism Theorem - The first isomorphism theorem
• Second Isomorphism Theorem - The second isomorphism theorem
• Third Isomorphism Theorem - The third isomorphism theorem
• Composition Series - Composition series and Jordan-Hölder theorem
• Simple Groups - Definition and examples of simple groups

4 - Jordan-Hölder Theorem

• Jordan-Hölder Theorem - The Jordan-Hölder theorem and its significance
• Alternating Groups - The alternating groups and their properties
• Direct Products - Direct products of groups

5 - Group Actions

• Group Actions - Definition and examples of group actions
• Semi-direct Products - Semi-direct products and their construction

6 - Orbits and Stabilizers

• Orbits and Stabilizers - Orbits, stabilizers, and the orbit-stabilizer theorem
• Orbits - Definition and properties of orbits
• Stabilizers - Definition and properties of stabilizers
• Orbit-Stabilizer Theorem - The orbit-stabilizer theorem
• Class Equation - The class equation and its applications
• Burnside's Lemma - Burnside's lemma for counting orbits

7 - Sylow's Theorems

• Sylow's Theorems - Overview of Sylow's theorems
• First Sylow Theorem - The first Sylow theorem
• Second Sylow Theorem - The second Sylow theorem
• Third Sylow Theorem - The third Sylow theorem

8 - Commutator Subgroups

• Commutator Subgroups - Definition and properties of commutator subgroups
• Solvable Groups - Solvable groups and their properties
• Nilpotent Groups - Nilpotent groups and their properties

🗂️ Part II: Ring and Field Theory - Beyond a Single Operation

9 - Rings and Ideals

• Rings and Ideals - Ring axioms, ideals, quotient rings, ring homomorphisms
• Rings - Definition and basic properties of rings
• Ideals - Definition and properties of ideals
• Quotient Rings - Construction and properties of quotient rings
• Ring Homomorphisms - Ring homomorphisms and their properties

10 - Special Ideals and Domains

• Special Ideals and Domains - Prime ideals, maximal ideals, Euclidean domains, PIDs
• Prime and Maximal Ideals - Prime and maximal ideals
• Principal Ideal Domains - Principal ideal domains (PIDs)
• Euclidean Domains - Euclidean domains and the division algorithm
• Chinese Remainder Theorem - The Chinese remainder theorem

11 - Unique Factorization Domains

• Unique Factorization Domains - UFDs and unique factorization
• Gauss's Lemma - Gauss's lemma for polynomial rings
• Eisenstein's Criterion - Eisenstein's criterion for irreducibility

12 - Polynomial Rings

• Polynomial Rings - Polynomial rings, Eisenstein's criterion, irreducibility

13 - Modules over PID

• Modules over PID - Modules, fundamental theorem, Jordan canonical form

🗂️ Part III: Galois Theory - The Symmetry of Equations

14 - Field Extensions

• Field Extensions - Field extensions, degree, algebraic vs transcendental

15 - Normal Extensions

• Normal Extensions - Normal extensions, splitting fields

16 - Separable Extensions

• Separable Extensions - Separable extensions, finite fields

17 - Galois Theory I

• Galois Theory I - Galois groups, fundamental theorem

18 - Galois Theory II

• Galois Theory II - Proof of fundamental theorem

19 - Solving Polynomials

• Solving Polynomials - Solvability by radicals, insolvability of quintic

20 - Infinite Galois Theory

• Infinite Galois Theory - Infinite extensions, profinite groups

21 - Algebraic Closures

• Algebraic Closures - Algebraic closures, transcendence basis

22 - Noether Normalization

• Noether Normalization - Noether normalization, Hilbert's Nullstellensatz

📁 Additional Resources

Reference Materials

• Key Definitions - Comprehensive glossary of all key terms

Supporting Folders

• Examples/ - Worked examples and applications
• Exercises/ - Practice problems and solutions
• Theorems/ - Important theorems and their proofs
• Applications/ - Real-world applications and connections

🎯 Learning Path

Beginner Level

1. Start with Basic Structures to understand the motivation
2. Study Groups and Subgroups for fundamental concepts
3. Learn Cosets and Lagrange's Theorem for group structure
4. Master Isomorphism Theorems for group relationships

Intermediate Level

1. Explore Rings and Ideals for algebraic structures
2. Study Field Extensions for field theory
3. Learn Galois Theory I for the fundamental theorem
4. Practice with examples and exercises

Advanced Level

1. Master Sylow's Theorems for finite group structure
2. Study Modules over PID for linear algebra connections
3. Explore Infinite Galois Theory for advanced topics
4. Apply to Algebraic Geometry and Number Theory

🔗 Quick Navigation

• README - Main course guide
• Key Definitions - All definitions in one place
• Examples/ - Worked examples
• Exercises/ - Practice problems
• Theorems/ - All theorems and proofs

This table of contents provides a complete overview of the Abstract Algebra course structure. Follow the numbered order for systematic learning, or use the quick navigation for specific topics.