🔗 Abstract Algebra
Based on Tom Judson's Abstract Algebra: Theory and Applications, licensed GFDL.
The structures underneath category theory: groups, rings, fields. Runnable Python, plain English, diagrams.
| Chapter | |||
|---|---|---|---|
| 1. | Groups | A set with one operation that is closed, associative, has an identity, and has inverses | 🔗 |
| 2. | Subgroups | Cyclic groups generated by one element, and two tests for when a subset is itself a group | 🔗 |
| 3. | Homomorphisms | Functions between groups that preserve the operation, and the kernel that measures what they forget | 🔗 |
| 4. | Isomorphisms | A bijective homomorphism says two groups are the same structure wearing different labels | 🔗 |
| 5. | Cosets and Lagrange's Theorem | Subgroups partition a group into equal-sized cosets, so subgroup order divides group order | 🔗 |
| 6. | Rings | Two operations, addition and multiplication, with ideals playing the role of normal subgroups | 🔗 |
| 7. | Integral Domains and Fields | Rings where cancellation works, and fields where every nonzero element has a multiplicative inverse | 🔗 |
| 8. | Lattices and Boolean Algebras | Partial orders with meets and joins, connecting algebra to logic and set theory | 🔗 |
📺 Video lectures: MIT 18.703 Modern Algebra
Neighbors
- 🐱 Category Theory — groups, rings, and monoids reappear as objects in a category
- # Number Theory — modular arithmetic is the integers mod n
- 🔐 Cryptography — RSA and elliptic curves are applied abstract algebra
- 📐 Linear Algebra — vector spaces are a special case of modules