Algebra

Self-study notes on MacLane and Birkhoff’s Algebra. These notes cover topics roughly equivalent to a one-semester undergraduate course on group and ring theory.

One of my favorite textbooks in algebra, Algebra offers a structured and cohesive exposition by connects algebraic constructions (such as quotients, kernels, and freely generated objects) through the language of category theory.

Earlier notes are hosted by notion; click on the collapsible headings to expand contents.

1. Groups (Ch. 2, 7): isomorphism theorems, Sylow theorems, the Jordan-Holder theorem, simplicity of alternating groups.
2. Rings (Ch. 3): ideals, polynomials and unique factorization, integral domains and fields.
3. Universal constructions (Ch. 4, 15): categorical aspects of algebraic structures, focusing on universality and functors. Hom-functors and representations, contravariance and duality.
4. Multilinear algebra (Ch. 16) is covered in miscellaneous notes.