Abstract Algebra
Groups, rings, fields, and Galois theory — the structural lens on mathematics.
01Abstract Algebra (1): Groups — Your First Encounter with Algebraic Structure
From integers to symmetries, we build the formal definition of a group, prove Lagrange's theorem, and compute our first …
02Abstract Algebra (2): Group Actions — How Groups Move Things Around
We formalize how groups act on sets, prove the orbit-stabilizer theorem, derive Burnside's lemma, and count necklaces.
03Abstract Algebra (3): Quotient Groups and Homomorphisms: The Art of Collapsing Structure
Normal subgroups, quotient constructions, and the isomorphism theorems — how to systematically simplify groups while …
04Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups
The Sylow theorems give us a systematic way to find and count subgroups of prime-power order — the sharpest tool for …
05Abstract Algebra (5): Rings and Ideals — When Multiplication Enters the Picture
Adding multiplication to the mix: rings, integral domains, ideals, and quotient rings — the algebraic structures behind …
06Abstract Algebra (6): Polynomial Rings — Factorization and Unique Decomposition
The division algorithm, irreducibility tests, and the climb from Z to Z[x] to Q[x] — understanding when and why unique …
07Abstract Algebra (7): Field Extensions — Building Bigger Number Systems
Algebraic and transcendental extensions, the tower law, minimal polynomials, and splitting fields — the machinery that …
08Abstract Algebra (8): Galois Theory — The Bridge Between Fields and Groups
The Fundamental Theorem of Galois Theory establishes a perfect correspondence between intermediate fields and subgroups …
09Abstract Algebra (9): Modules — Generalizing Vector Spaces
Modules over rings generalize vector spaces over fields — the structure theorem for finitely generated modules over PIDs …
10Abstract Algebra (10): Representation Theory — Groups Acting on Vector Spaces
Representing abstract groups as matrices makes them concrete and computable — Maschke's theorem, Schur's lemma, and …
11Abstract Algebra (11): Category Theory — The Language of Mathematical Structure
Categories, functors, and natural transformations provide a universal language for mathematical structure — and …
12Abstract Algebra (12): Algebra in the Wild — Cryptography, Coding Theory, and Beyond
From RSA encryption to error-correcting codes to particle physics — abstract algebra's most powerful real-world …