Abstract Algebra

Contents

1. Groups
2. Subgroups
3. Lagrange”s Theorem
4. Normal Subgroups and Quotient Groups
5. Homomorphisms and Isomorphism Theorems
6. Group Actions
7. The Sylow Theorems
8. Rings
9. Ideals and Quotient Rings
10. Polynomial Rings
11. Euclidean Domains, PIDs, and UFDs
12. Field Theory
13. Galois Theory Fundamentals
14. Additional Results
15. Worked Examples
16. Classification of Groups of Small Order
17. Common Pitfalls
18. Problem Set
19. Summary of Key Results

Overview

University-level abstract algebra notes covering groups, rings, fields, and Galois theory.

Topics Covered

• Groups and Subgroups: Definitions, examples, Lagrange’s theorem
• Homomorphisms: Isomorphism theorems, group actions, Sylow theorems
• Rings and Ideals: Polynomial rings, Euclidean domains, PIDs, UFDs
• Field Theory: Extensions, splitting fields, Galois theory

Prerequisites

• Mathematical proofs and logic
• Basic linear algebra (helpful but not required)
• Mathematical maturity

How to Use These Notes

Start with groups to build foundational knowledge, then progress to rings and fields. Each section includes worked examples and practice problems.

Navigation

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

• Detailed explanations of key concepts
• Worked examples with step-by-step solutions
• Practice problems with answers
• Common pitfalls and how to avoid them
• Connections to other areas of mathematics

Study Tips

1. Master the definitions: Abstract algebra requires precise understanding of definitions
2. Practise proofs: Learn to write clear, rigorous proofs
3. Draw Cayley tables: Visualise group structure for small examples
4. Learn standard examples: Know the properties of common groups (cyclic, symmetric, dihedral)
5. Connect to applications: Relate abstract concepts to number theory, geometry, and physics