Abstract Algebra Course

Lesson 1 • Binary Operations and Properties

Defines binary operations and examines closure, associativity, commutativity, and identity. Connects operational properties to the axioms of groups and rings.

Lesson 2 • Introduction to Mathematical Proof

Covers direct proof, contrapositive, contradiction, and induction. Equips students to construct and evaluate rigorous algebraic arguments throughout the course.

Lesson 3 • Sets, Relations, and Functions

Reviews set-theoretic notation, equivalence relations, and function types. Provides the logical scaffolding for defining algebraic structures rigorously.

Lesson 1 • Cosets and Lagrange's Theorem

Develops left and right cosets, proves Lagrange's theorem, and derives index. Establishes the fundamental constraint on subgroup orders in finite groups.

Lesson 2 • Group Homomorphisms

Defines homomorphisms, kernels, and images, and proves basic properties. Introduces the morphism concept that unifies all algebraic structures studied later.

Lesson 3 • Definition and Examples of Groups

States the group axioms and surveys canonical examples including integers, symmetries, and matrix groups. Grounds abstract definitions in concrete, familiar settings.

Lesson 4 • Cyclic Groups and Their Structure

Classifies cyclic groups up to isomorphism and analyses their subgroup lattice. Provides the first complete structural classification result in the course.

Lesson 1 • The Isomorphism Theorems

States and proves the first, second, and third isomorphism theorems. Provides the main tools for comparing group structures and simplifying homomorphism analysis.

Lesson 2 • Quotient Group Construction

Constructs the quotient group G/N and verifies the group axioms. Demonstrates how factoring by a normal subgroup creates a new, simpler group.

Lesson 3 • Simple Groups and Composition Series

Defines simple groups and composition series, and states the Jordan-Hölder theorem. Introduces the concept of irreducible building blocks of finite groups.

Lesson 1 • Group Actions on Sets

Defines group actions, orbits, and stabilisers, and proves the orbit-stabiliser theorem. Unifies conjugation, coset multiplication, and symmetry as instances of one framework.

Lesson 2 • Burnside's Lemma and Counting

Applies Burnside's lemma to count orbits in combinatorial settings. Demonstrates the power of group actions in solving enumeration problems.

Lesson 3 • Conjugation Action and Class Equation

Analyses conjugation as a group action and derives the class equation. Uses the class equation to prove properties of p-groups.

Lesson 4 • Classification of Groups of Small Order

Applies Sylow theory to classify groups of orders up to 30. Synthesises all group theory tools into a systematic classification methodology.

Lesson 1 • Integral Domains and Fields

Characterises integral domains, fields, and the field of fractions construction. Prepares students for polynomial rings and field extensions in later chapters.

Lesson 2 • Rings: Axioms and Basic Examples

Defines rings, commutative rings, and rings with unity, surveying integers, polynomials, and matrices. Establishes the two-operation framework that extends group theory.

Lesson 3 • Prime and Maximal Ideals

Defines prime and maximal ideals and characterises them via quotient ring properties. Connects ideal types to integral domains and fields.

Lesson 4 • Quotient Rings and Homomorphisms

Constructs quotient rings R/I and proves the ring isomorphism theorems. Parallels the group quotient construction and extends the isomorphism theorem framework.

Lesson 1 • Unique Factorisation Domains

Proves every PID is a UFD and analyses irreducibles vs. primes. Establishes the algebraic analog of the fundamental theorem of arithmetic.

Lesson 2 • Euclidean Domains and PIDs

Defines Euclidean domains and principal ideal domains and proves the inclusion chain ED⊂PID⊂UFD. Provides the structural hierarchy for factorisation theory.

Lesson 3 • Irreducibility Criteria

Develops tests for irreducibility including rational root, Eisenstein, and reduction mod p. Equips students to determine whether a polynomial factors over a given field.

Lesson 4 • Gauss's Lemma and Applications

Proves Gauss's lemma on primitive polynomials and applies it to factorisation over Z vs. Q. Connects integer and rational polynomial factorisation.

Lesson 1 • Field Extensions: Basics

Defines field extensions, algebraic and transcendental elements, and the degree [E:F]. Introduces the tower law as the key multiplicative tool for extension degrees.

Lesson 2 • Solvability by Radicals

Defines radical extensions and solvable groups, then proves the Abel-Ruffini theorem. Demonstrates why degree-5 polynomials are generally unsolvable by radicals.

Lesson 3 • Splitting Fields and Algebraic Closure

Constructs splitting fields for polynomials and states the existence of algebraic closures. Provides the setting in which all roots of a polynomial coexist.

Lesson 4 • The Galois Group and Correspondence

Defines the Galois group Gal(E/F) and proves the fundamental theorem of Galois theory. Establishes the bijection between intermediate fields and subgroups of the Galois group.

Lesson 1 • Applications to Abelian Groups

Applies the structure theorem to classify all finitely generated abelian groups. Demonstrates the theorem's power as a complete classification tool.

Lesson 2 • Modules Over Rings

Defines left R-modules, submodules, and module homomorphisms. Positions modules as the common generalisation of abelian groups and vector spaces.

Lesson 3 • Structure Theorem for Modules Over PIDs

Proves the structure theorem for finitely generated modules over PIDs in invariant factor and elementary divisor forms. Unifies classification of abelian groups and canonical forms.

Lesson 4 • Exact Sequences and Projective Modules

Introduces short exact sequences, split sequences, and projective modules. Provides the language for describing module extensions and direct sum decompositions.