Algebra

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About This Series

Algebra begins with "computation using variables" and develops into the abstract study of the structure of numbers. This series progresses from high-school "numbers and expressions" through groups, rings, fields, Galois theory, and on to homological algebra.

Algebra underpins every branch of mathematics and is widely applied in physics, engineering, and computer science.

Learning by Level

Introduction

High-school level

  • Algebraic expressions and polynomials
  • Expansion and factoring
  • Quadratic equations and functions
  • Complex numbers and higher-degree equations

Elementary

Undergraduate 1st–2nd year

  • Basics of groups
  • Group homomorphisms
  • Rings and fields
  • Polynomial rings
  • Field extensions

Intermediate

Undergraduate 3rd–4th year

  • Galois theory
  • Solvability of equations
  • Finite fields
  • Basics of modules

Advanced

Graduate level

  • Homological algebra
  • Representation theory
  • Commutative ring theory
  • Category theory and algebra

Learning Path

Intro High school Elementary Undergrad 1–2 Intermediate Undergrad 3–4 Advanced Graduate Intro: expansion, factoring, quadratics, complex numbers Elementary: groups, rings, fields, polynomial rings, extensions Intermediate: Galois theory, solvability, finite fields, modules Advanced: homological algebra, representations, commutative rings

Main Topics

Polynomial Arithmetic

Expansion, factoring, polynomial division — the foundational techniques of algebraic computation.

Equations

Methods and theory for solving equations from linear to higher-degree.

Algebraic Structures

Abstract understanding of mathematical structures: groups, rings, and fields.

Galois Theory

Elucidating the solvability of equations through the correspondence between field extensions and groups.