Abstract Algebra Knowledge Base

This is a comprehensive knowledge base for Abstract Algebra, organized into three main parts following the standard curriculum structure. All mathematical content is formatted with LaTeX for proper rendering in Obsidian.

📚 Overview

Abstract Algebra is the study of algebraic structures and their properties. This knowledge base 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

🗂️ Directory Structure

Part I: Group Theory - The Mathematics of Symmetry

• 0 - Basic Structures in Algebra/
  • Basic Structures.md - Sets of numbers, vector spaces, algebraic structures
• 1 - Groups and Subgroups/
  • Groups and Subgroups.md - Group axioms, subgroups, dihedral groups, permutation groups
• 2 - Cosets and Lagrange's Theorem/
  • Cosets and Lagrange's Theorem.md - Cosets, Lagrange's theorem, normal subgroups, quotient groups
• 3 - Isomorphism Theorems/
  • Isomorphism Theorems.md - Three isomorphism theorems, composition series, simple groups
• 4 - Jordan-Hölder Theorem/
  • Jordan-Hölder theorem, alternating groups, direct products
• 5 - Group Actions/
  • Group actions, semi-direct products
• 6 - Orbits and Stabilizers/
  • Orbits, stabilizers, class equation
• 7 - Sylow's Theorems/
  • Sylow's theorems, prime-power subgroups
• 8 - Commutator Subgroups/
  • Commutator subgroups, solvable groups, nilpotent groups

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

• 9 - Rings and Ideals/
  • Rings and Ideals.md - Ring axioms, ideals, quotient rings, ring homomorphisms
• 10 - Special Ideals and Domains/
  • Prime ideals, maximal ideals, Euclidean domains, PIDs
• 11 - Unique Factorization Domains/
  • UFDs, Gaussian integers, factorization
• 12 - Polynomial Rings/
  • Polynomial rings, Eisenstein's criterion, irreducibility
• 13 - Modules over PID/
  • Modules, fundamental theorem, Jordan canonical form

Part III: Galois Theory - The Symmetry of Equations

• 14 - Field Extensions/
  • Field extensions, degree, algebraic vs transcendental
• 15 - Normal Extensions/
  • Normal extensions, splitting fields
• 16 - Separable Extensions/
  • Separable extensions, finite fields
• 17 - Galois Theory I/
  • Galois Theory I.md - Galois groups, fundamental theorem
• 18 - Galois Theory II/
  • Proof of fundamental theorem
• 19 - Solving Polynomials/
  • Solvability by radicals, insolvability of quintic
• 20 - Infinite Galois Theory/
  • Infinite extensions, profinite groups
• 21 - Algebraic Closures/
  • Algebraic closures, transcendence basis
• 22 - Noether Normalization/
  • Noether normalization, Hilbert's Nullstellensatz

📁 Additional Folders

• Definitions/
  • Key Definitions.md - Comprehensive glossary of all key terms
• Theorems/ - Important theorems and their proofs
• Examples/ - Worked examples and applications
• Exercises/ - Practice problems and solutions
• Applications/ - Real-world applications and connections

🎯 How to Use This Knowledge Base

1. Start with Part I - Group theory provides the foundation
2. Follow the numbered order - Each section builds on previous ones
3. Use the additional folders - For quick reference and practice
4. Cross-reference - Many concepts appear in multiple contexts

🔗 Quick Navigation

• Abstract-Algebra.md - Main course outline
• Definitions/Key Definitions.md - All definitions in one place
• Theorems/ - All theorems and proofs
• Examples/ - Worked examples
• Exercises/ - Practice problems

📝 Note-Taking Tips

• Create notes in each numbered folder as you study
• Use the Definitions/ folder for quick reference
• Add examples to the Examples/ folder
• Practice with exercises in the Exercises/ folder
• Connect related concepts across different parts

🧮 LaTeX Formatting

All mathematical expressions in this knowledge base are formatted using LaTeX syntax:

• Inline math: $x^2 + y^2 = z^2$ renders as $x^2+y^2=z^2$
• Display math: $$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$ renders as block equations
• Greek letters: $\alpha, \beta, \gamma, \delta$ render as $\alpha ,\beta ,\gamma ,\delta$
• Special symbols: $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ render as $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$

📚 Key Mathematical Notation

Sets and Numbers

• $\mathbb{Z}$ - Integers
• $\mathbb{Q}$ - Rational numbers
• $\mathbb{R}$ - Real numbers
• $\mathbb{C}$ - Complex numbers
• ${\mathbb{Z}}_n$ - Integers modulo n

Groups

• $G,H$ - Groups
• $|G|$ - Order of group G
• $H\le G$ - H is a subgroup of G
• $N⊴G$ - N is a normal subgroup of G
• $G/N$ - Quotient group
• $D_n$ - Dihedral group of order 2n
• $S_n$ - Symmetric group on n letters
• $A_n$ - Alternating group on n letters

Rings and Fields

• $R,S$ - Rings
• $K,L,E$ - Fields
• $R/I$ - Quotient ring
• $K[x]$ - Polynomial ring over K
• $[L:K]$ - Degree of field extension

Galois Theory

• $\text{Gal}(L/K)$ - Galois group of extension L/K
• $L^H$ - Fixed field of subgroup H

🎓 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

🔍 Search and Navigation

Use Obsidian's search features to find specific topics:

• Tags: Use #group-theory, #ring-theory, #galois-theory
• Links: Follow internal links with [[filename]]
• Graph view: Visualize connections between concepts

📖 Recommended Reading

Primary Sources

• Dummit & Foote, "Abstract Algebra"
• Lang, "Algebra"
• Artin, "Algebra"

Supplementary Materials

• Rotman, "An Introduction to the Theory of Groups"
• Hungerford, "Algebra"
• Jacobson, "Basic Algebra"

This knowledge base follows the standard Abstract Algebra curriculum and can be used for self-study or as a reference for coursework. All mathematical content is properly formatted with LaTeX for optimal rendering in Obsidian.