Abstract Algebra | Mathematics

sources:

• text: Spivak - Calculus

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

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

• Groups and Subgroups: Definitions, examples, Lagrange’s theorem. Groups capture symmetry; subgroups are subsets closed under the group operation.
• Homomorphisms: Isomorphism theorems, group actions, Sylow theorems. Homomorphisms preserve structure; the isomorphism theorems describe quotient structures.
• Rings and Ideals: Polynomial rings, Euclidean domains, PIDs, UFDs. Rings extend groups with a second operation; ideals enable quotient rings.
• Field Theory: Extensions, splitting fields, Galois theory. Fields are the most structured algebraic objects; Galois theory connects field extensions to polynomial solvability.

• Mathematical proofs and logic. Abstract algebra is proof-based — you will write many proofs about abstract structures.
• Basic linear algebra (helpful but not required). Vector spaces are examples of modules; eigenvalues connect to group representations.
• Mathematical maturity. You should be comfortable with abstraction and working with definitions.

Start with groups to build foundational knowledge, then progress to rings and fields. Each section includes worked examples and practice problems. The key is to work through proofs yourself — reading proofs is not enough.

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

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

Abstract algebra distils the essence of arithmetic into algebraic structures. A group captures the idea of symmetry: any set of operations that can be composed and undone forms a group, from the rotations of a square to the permutations of a Rubik’s cube. A ring extends this by adding a second operation, like multiplication, enabling the study of number systems and polynomial equations. A field adds the requirement that every non-zero element has a multiplicative inverse, producing the familiar arithmetic of fractions. The power of abstraction is that the same theorems apply to wildly different objects: the structure of finite groups illuminates crystal symmetries, Galois theory connects field extensions to polynomial solvability, and ring theory underpins modern cryptography.

1. Master the definitions: Abstract algebra requires precise understanding of definitions. Misremembering a definition leads to incorrect proofs.
2. Practise proofs: Learn to write clear, rigorous proofs. Abstract algebra is excellent training for mathematical reasoning.
3. Draw Cayley tables: Visualise group structure for small examples. Cayley tables reveal patterns in multiplication.
4. Learn standard examples: Know the properties of common groups (cyclic, symmetric, dihedral). These serve as test cases for general theorems.
5. Connect to applications: Relate abstract concepts to number theory, geometry, and physics. Applications provide motivation.

• Linear Algebra: Vector spaces and linear transformations; vector spaces are modules over fields.

• Number Theory: Group theory in modular arithmetic; (Z/nZ)* is a group under multiplication.

• Topology: Topological groups and algebraic topology; group theory underpins homology groups.

• Quantum Mechanics

• Graph Theory