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
  5. Polynomials on Steroids

Fields

  1. Extended Field Trip
  2. The Algebraic Closure
  3. Revisiting Automorphisms [TBC]
  4. Separable Extensions [TBC]

Galois Theory

  1. Galois Extensions [TBC]
  2. The Galois Group [TBC]
  3. Symmetric Functions [TBC]
  4. Quintic Insolvability [TBC]

Exercises

Groups

  1. Euler’s Coprimes
  2. Sylow Theorems [TBC]

Rings

  1. Ring Isomorphisms
  2. Chinese Mathematics
  3. Generalised Fractions

Fields

  1. Euclidean Constructibility [TBC]
  2. Finite Field Arithmetic [TBC]

Galois Theory

  1. Cyclotomic Polynomials [TBC]