Abstract Algebra

Symmetry, structure, and the objects that formalise them

About this course

Groups, rings and fields developed from symmetry — quotients, homomorphisms, factorisation and field extensions, ending at finite fields.

What you will be able to do

  • Work with groups, subgroups and element orders, and use Lagrange's theorem
  • Construct quotient groups and apply the first isomorphism theorem
  • Use group actions to count
  • Work in rings and polynomial rings, including ideals and quotients
  • Construct field extensions and describe the finite fields

What this course covers

  • Symmetry as a rigid transformation
  • Representing symmetries as permutations
  • Composition of symmetries and non-commutativity
  • The four group axioms
  • The composition table of the triangle group
  • Inverses and element order
  • Worked calculation: simplifying RSR
  • Representatives and coset equality criterion
  • Coset definition and notation
  • Left cosets partition the group
  • Bijection from subgroup to each coset
  • Element order divides group order
  • Lagrange's theorem
  • Groups of prime order are cyclic
  • Euclidean domains and the implication chain overview
  • Units, associates, irreducibles, and prime elements
  • Every ideal in a Euclidean domain is principal
  • Ascending chain condition and existence of irreducible factorization
  • Irreducibles are prime in a PID
  • Uniqueness of irreducible factorization in a PID
  • Failure of unique factorization in ℤ[√−5]
  • Ring definition: two operations linked by distributivity
  • Units and multiplicative inverses
  • Zero divisors and failure of cancellation
  • Cancellation vs. division: two distinct questions
  • Integral domains and fields: definitions and relationship
  • Every finite integral domain is a field
  • Multiplication maps as a unifying diagnostic tool
  • Coset multiplication and well-definedness problem
  • Left cosets, right cosets, and conjugation
  • Normal subgroups and the normality condition
  • Normality as necessary and sufficient for well-defined coset product
  • Quotient group construction and group laws
  • Additive quotient groups and clock arithmetic
  • Quotient construction summary and projection to homomorphisms
  • Division with Remainder in the Integers
  • The Euclidean Algorithm and Its Correctness
  • Bézout's Identity and the Extended Euclidean Algorithm
  • Units and Modular Inverses via Coprimality
  • Cancellation and Non-Units in Modular Arithmetic
  • Linear Congruences: Existence, Count, and Solution Method
  • Summary: Structured Solution Method for Congruences
  • Field extensions and vector space structure
  • Adjoining a square root: basis, closure, and inverses
  • Tower law: degrees multiply through a product basis
  • Minimal polynomial of an algebraic element
  • Evaluation map, kernel, and quotient ring isomorphism
  • Constructing a four-element field via an irreducible quadratic
  • Changing the generator without changing the field
  • Group definition and the operation as part of the example
  • Non-examples, counterexamples, and the limits of cancellation
  • Uniqueness of identity and inverses
  • Cancellation and equation solving in groups
  • Subgroups and the inherited operation
  • The subgroup test (ab⁻¹ criterion)
  • Intersections, unions, and the finite closure theorem
  • Finite fields: definition and prime subfield
  • Prime-power size theorem for finite fields
  • General classification theorem (stated, not proved)
  • Constructing F4 as a polynomial quotient
  • Cyclic multiplicative group of a finite field
  • Formal polynomials vs. induced functions on finite fields
  • Frobenius automorphism
  • Modular arithmetic on a clock group
  • Generated subgroups from a single element
  • Element order and the first positive return
  • Generator criterion: gcd equals one
  • Order formula via GCD: modulus divided by gcd
  • Infinite cyclic groups and infinite element order
  • Unified summary: repetition, order, and subgroup structure
  • Polynomials as formal coefficient arrays
  • Polynomial ring arithmetic and the degree formula
  • Polynomial division algorithm over a field
  • Remainder theorem and factor theorem
  • Euclidean algorithm, GCD, and Bézout identity for polynomials
  • Irreducibility and its dependence on the coefficient field
  • Root bound and polynomial-vs-function distinction
  • Homomorphisms and the operation-square condition
  • Kernel and image as subgroups
  • Fibers as cosets of the kernel
  • First Isomorphism Theorem
  • Collisions vs. missed targets: image vs. codomain
  • Applications: parity map and integer examples
  • Well-definedness and the representative trap
  • Permutations as bijections and the symmetric group
  • Cycle notation and disjoint cycle decomposition
  • Composition of permutations with overlapping cycles
  • Order of a permutation via least common multiple
  • Inverses and factoring cycles into transpositions
  • Inversion count and well-defined parity of permutations
  • The alternating group A_n and its properties
  • Group actions: definition and axioms
  • Transitivity and faithfulness of actions
  • Orbits and stabilizers
  • Orbit-stabilizer theorem
  • Burnside's lemma: fixed-pair double count
  • Cycle structure and fixed coloring counts
  • Dependence of orbit count on the specified action
  • Why quotient rings need absorption
  • Ideals and the absorption condition
  • Principal ideals
  • Quotient ring construction and well-definedness
  • Polynomial division as canonical representatives
  • Prime and maximal ideals via quotient rings
  • Evaluation map and its kernel as a principal ideal

Syllabus

  1. Groups and Their Structure

    1. L1 · What a Symmetry Actually Is
    2. L2 · Groups, Subgroups and the Axioms at Work
    3. L3 · Cyclic Groups and Element Orders
    4. L4 · Permutations and Their Structure

  2. Cosets, Quotients and Homomorphisms

    1. L5 · Cosets and Lagrange’s Theorem
    2. L6 · Normal Subgroups and Quotients
    3. L7 · Homomorphisms and the First Isomorphism Theorem
    4. L8 · Group Actions and Counting

  3. Rings

    1. L9 · Rings, Units and Zero Divisors
    2. L10 · Euclid, Bézout and Modular Inverses
    3. L11 · Polynomial Rings
    4. L12 · Ideals and Quotient Rings

  4. Factorisation and Fields

    1. L13 · Factorization
    2. L14 · Field Extensions
    3. L15 · Finite Fields and the Course Synthesis