Groups and Fields· Study notes
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.
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
Read the notes as a guided lesson
Walk the notes one box at a time — prose, worked examples, quick checks — with a timed practice round after every section. For the full course with interactive lessons, go back to the course.
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
A Gallery of Groups
6 lessonsIntegers mod n, units mod n, symmetries of polygons, and matrix groups
Subgroups and Cyclic Groups
5 lessonsSubgroup tests, generators, and the complete structure of cyclic groups
Permutation Groups
6 lessonsCycle notation, composing permutations, order, parity, and Cayley's theorem
Cosets and Lagrange's Theorem
5 lessonsPartitioning a group into cosets, and what the size of a group forces
Homomorphisms and Isomorphisms
6 lessonsStructure-preserving maps, kernels and images, and telling groups apart
Normal Subgroups and Quotient Groups
6 lessonsWhen cosets form a group, and the three isomorphism theorems
Direct Products and Finite Abelian Groups
6 lessonsBuilding groups from smaller ones, and classifying every finite abelian group
Group Actions
6 lessonsGroups acting on sets, orbits and stabilizers, and counting with Burnside's lemma
Conjugacy and the Class Equation
5 lessonsConjugacy classes, the class equation, and why p-groups have a center
The Sylow Theorems
6 lessonsSylow subgroups, counting them, and classifying groups of small order
Rings and Fields
5 lessonsTwo operations at once: rings, fields, integral domains and the characteristic
Polynomials over a Field
6 lessonsDivision, roots, irreducibility tests and unique factorization in F[x]
Field Extensions
5 lessonsNew fields from irreducible polynomials, their degrees, and the tower law
Splitting Fields and Constructions
5 lessonsFields that contain every root, cyclotomic polynomials, and the ancient construction problems
Finite Fields
5 lessonsFields with p^n elements: their sizes, cyclic multiplicative groups, subfields and irreducible polynomials
Galois Theory
5 lessonsWhere the two halves meet: symmetry groups of fields, the fundamental theorem, and the unsolvable quintic