← All courses G G G Algebra
Abstract algebra in two halves. Groups: the axioms, cyclic and permutation groups, Lagrange's theorem, homomorphisms and quotients, finite abelian groups, group actions, conjugacy and the Sylow theorems. Fields: rings and fields, polynomials and irreducibility, field extensions and the tower law, splitting fields and ruler-and-compass constructions, finite fields, and Galois theory.
Before this: Proof Writing & Problem Solving , Linear Algebra What you'll be able to do
Verify the group axioms and compute in Z_n, U(n), D_n and S_n Use Lagrange, orbit-stabilizer and the class equation to pin down orders Build quotient groups and apply the isomorphism theorems Classify finite abelian groups and groups of small order with Sylow Test polynomials for irreducibility and compute degrees of field extensions Build finite fields and compute Galois groups of small polynomials Start the course
Go through each topic step by step — prose, worked examples, quick checks, and a timed practice round after every section.
Every test ends with a score by topic and a list of what to revisit. Test out of a topic with 80% or better to mark it mastered.
Topics 1. Groups and the Group Axioms
6 lessons Binary operations, the four axioms, and the first theorems they imply
2. A Gallery of Groups
6 lessons Integers mod n, units mod n, symmetries of polygons, and matrix groups
3. Subgroups and Cyclic Groups
5 lessons Subgroup tests, generators, and the complete structure of cyclic groups
4. Permutation Groups
6 lessons Cycle notation, composing permutations, order, parity, and Cayley's theorem
5. Cosets and Lagrange's Theorem
5 lessons Partitioning a group into cosets, and what the size of a group forces
6. Homomorphisms and Isomorphisms
6 lessons Structure-preserving maps, kernels and images, and telling groups apart
7. Normal Subgroups and Quotient Groups
6 lessons When cosets form a group, and the three isomorphism theorems
8. Direct Products and Finite Abelian Groups
6 lessons Building groups from smaller ones, and classifying every finite abelian group
Groups acting on sets, orbits and stabilizers, and counting with Burnside's lemma
10. Conjugacy and the Class Equation
5 lessons Conjugacy classes, the class equation, and why p-groups have a center
11. The Sylow Theorems
6 lessons Sylow subgroups, counting them, and classifying groups of small order
12. Rings and Fields
5 lessons Two operations at once: rings, fields, integral domains and the characteristic
13. Polynomials over a Field
6 lessons Division, roots, irreducibility tests and unique factorization in F[x]
14. Field Extensions
5 lessons New fields from irreducible polynomials, their degrees, and the tower law
15. Splitting Fields and Constructions
5 lessons Fields that contain every root, cyclotomic polynomials, and the ancient construction problems
16. Finite Fields
5 lessons Fields with p^n elements: their sizes, cyclic multiplicative groups, subfields and irreducible polynomials
17. Galois Theory
5 lessons Where the two halves meet: symmetry groups of fields, the fundamental theorem, and the unsolvable quintic