Abstract-Algebra

Sep 23, 2021 Abstract Algebra 68 min read

Abstract 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 applications, and where to go next.

Sep 21, 2021 Abstract Algebra 62 min read

Abstract Algebra (11): Category Theory — The Language of Mathematical Structure

Categories, functors, and natural transformations provide a universal language for mathematical structure — and universal properties replace ad hoc constructions with elegant characterizations.

Sep 19, 2021 Abstract Algebra 62 min read

Abstract 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 character theory give us powerful classification tools.

Sep 17, 2021 Abstract Algebra 68 min read

Abstract Algebra (9): Modules — Generalizing Vector Spaces

Modules over rings generalize vector spaces over fields — the structure theorem for finitely generated modules over PIDs unifies the theory of abelian groups and canonical forms.

Sep 15, 2021 Abstract Algebra 70 min read

Abstract 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 — and settles the ancient question of solvability by radicals.

Sep 13, 2021 Abstract Algebra 66 min read

Abstract Algebra (7): Field Extensions — Building Bigger Number Systems

Algebraic and transcendental extensions, the tower law, minimal polynomials, and splitting fields — the machinery that makes Galois theory possible.

Sep 11, 2021 Abstract Algebra 70 min read

Abstract 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 factorization holds.

Sep 9, 2021 Abstract Algebra 68 min read

Abstract 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 number theory and polynomial arithmetic.

Sep 7, 2021 Abstract Algebra 66 min read

Abstract 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 classifying finite groups.

Sep 5, 2021 Abstract Algebra 70 min read

Abstract 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 preserving their essence.

Sep 3, 2021 Abstract Algebra 68 min read

Abstract 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.

Sep 1, 2021 Abstract Algebra 70 min read

Abstract 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 subgroup lattice.