School algebra is about solving for x. Abstract algebra is about something bolder: it forgets what the objects are — numbers, symmetries, shuffles of a deck, rotations of a cube — and studies only the rules they obey. Strip away everything except "you can combine two things to get a third, and here are the laws that combination follows", and a startling amount of mathematics falls out of almost nothing.

The pay-off is unification. The clock, the integers mod a prime, the symmetries of a triangle and the solutions of an equation turn out to be the same structure wearing different costumes. Prove a theorem once, at the level of the rules, and it holds everywhere the rules hold. This is the language of modern number theory, cryptography, particle physics and error-correcting codes.

The big idea: structure, not stuff

One thread runs through everything here. We fix a set and one or two operations on it, demand a short list of axioms (closure, associativity, an identity, inverses…), and then ask what must be true. Add more axioms and you climb a ladder — from a group (one operation) to a ring (two operations, add and multiply) to a field (where you can also divide). At each rung the objects get richer, and the theorems get sharper.

The shape of the journey

This course climbs that ladder in three stages.

• Stage A — Group Theory. One operation and its axioms: subgroups, cosets, quotients, homomorphisms, and the deep theorems about symmetry.
• Stage B — Ring Theory. Two operations at once — addition and multiplication — leading to ideals, polynomial rings and factorization.
• Stage C — Field & Galois Theory. Where division works too, culminating in Galois's stunning proof that the quintic has no formula.

Stage A — Group Theory

1. What Is a Group?
2. Examples of Groups
3. Subgroups
4. Cyclic Groups
5. Group Homomorphisms
6. Cosets and Lagrange's Theorem
7. Normal Subgroups and Quotient Groups
8. The Isomorphism Theorems
9. Permutation Groups
10. Group Actions and Orbits
11. Solvable and Simple Groups

Stage B — Ring Theory

1. What Is a Ring?
2. Integral Domains and Fields
3. Ring Homomorphisms
4. Ideals and Quotient Rings
5. Polynomial Rings
6. Irreducibility of Polynomials
7. Euclidean Domains and PIDs
8. Unique Factorization Domains

Stage C — Field & Galois Theory

1. Field Extensions
2. Algebraic and Transcendental Elements
3. The Degree of an Extension
4. Ruler-and-Compass Constructions
5. Splitting Fields
6. Finite Fields
7. The Galois Group
8. The Galois Correspondence
9. Solvability by Radicals
10. Insolvability of the Quintic

Let's get started

Everything begins with the simplest structure of all — a set with one well-behaved operation. Master the group and the rest of the ladder is just adding rules.

Let's get started → What Is a Group?