KindiakMath

Abstract Algebra

Here are my write-ups on the theory of abstract algebra and some non-trivial exercises therein. Abstract algebra seeks to formalise properties and structures common to many kinds of sets, and draw out some of their unified patterns. The goal of this set of posts is to introduce groups, rings, and fields, and if possible, touch on some baby Galois theory.

Theory

Groups

  1. Baby Group Theory
  2. Group Magic
  3. Finite Richness
  4. Group Isomorphisms
  5. The Language of Symmetry

Rings

  1. Law of the Rings
  2. Subrings and Ideals
  3. Commutative Ring Zoo
  4. Revisiting Factorisation

Fields

  1. Polynomials on Steroids
  2. Extended Field Trip
  3. The Algebraic Closure
  4. Revisiting Automorphisms

Galois Theory

  1. Galois Extensions
  2. Galois Par-Excellence
  3. Galois Group Theory
  4. Quintic Insolvability

Exercises

Groups

  1. Euler’s Coprimes
  2. Group Actions
  3. Sylow Theorems

Rings

  1. Ring Isomorphisms
  2. Chinese Mathematics
  3. Generalised Fractions
  4. Abelian Groups

Fields

  1. Euclidean Constructibility
  2. Finite Field Arithmetic

Galois Theory

  1. Cyclotomic Polynomials