Algebra
When does structure reveal more than quantity?
Rotations of a triangle, integers modulo seven, and the permutations of a set share no subject matter and obey the same four rules. Abstract algebra is the decision to study the rules and forget the objects, and the payoff is that one theorem then covers all of them.
Groups
What is the smallest useful amount of structure?
Four rules, and everything that obeys them
The film collects five objects that look unrelated, strips each to its operation, and finds the same four axioms every time. Nothing is defined until that has happened.
Dividing one structure by another
Quotient groups are the first genuinely difficult construction most people meet. The film builds one by hand before the word normal is used.
Rings
What happens when you keep multiplication but lose division?
Numbers you cannot always divide
The integers and the polynomials behave alike in a way that is not a coincidence, and the shared behaviour is what a ring is.
Unique factorisation, and where it breaks
Every integer factors into primes in one way only. In other rings it does not, and the failure is the origin of algebraic number theory.
Fields
Which equations can be solved at all?
Building a field to hold a root
If a polynomial has no root, invent one and see what the arithmetic forces. The film does this concretely before any theory appears.
Why the quintic has no formula
The impossibility of a general quintic formula is a statement about symmetry, not about algebra being hard. The document is mid-revision.
Representation theory
What if every group is secretly a group of matrices?
Algebra, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume VII ↓