---
title: Galois Theory
description: Galois theory connects field theory and group theory to study the roots of polynomial equations. Learners will understand how to determine the solvability of polynomials by radicals and solve classical geometric construction problems using algebraic techniques.
category: mathematics
subcategory: abstract-algebra
difficulty: beginner, intermediate, advanced
url: /subject/galois-theory
---

# Galois Theory

Galois theory connects field theory and group theory to study the roots of polynomial equations. Learners will understand how to determine the solvability of polynomials by radicals and solve classical geometric construction problems using algebraic techniques.

## Available Resources

1 Books • 2 Courses • 3 Websites

## Websites

### 1. This Week's Finds in Mathematical Physics, Week 201

**Author:** John Baez

John Baez's 2004 expository column introducing Galois theory through symmetry: Galois groups of field extensions, quadratic and cyclotomic number fields, the fundamental theorem, why the quintic has no radical solution, and classical construction problems, with pointers toward class field theory.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://math.ucr.edu/home/baez/week201.html

**Tags:** galois-theory, field-extensions, galois-groups, algebraic-number-theory, cyclotomic-fields

### 2. nLab Galois Theory

Entry in nLab, the research-level wiki for category theory and higher mathematics, presenting Galois theory as an equivalence between field extensions and Galois-set categories, plus Grothendieck's generalisation to fundamental groups and covering spaces. Readers see how the classical correspondence fits unifying categorical frameworks.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/Galois+theory

**Tags:** galois-theory, category-theory, grothendieck-galois-theory, field-extensions, abstract-algebra

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Galois Theory

Learn Galois Theory from Kevin Buzzard's engaging course. Explore field extensions, polynomial roots, and group theory applications!

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt

**Tags:** galois-theory, field-extensions, galois-groups, abstract-algebra, university-mathematics

### 2. Abstract Algebra

Learn Galois Theory with Benedict Gross's Abstract Algebra course. Explore field extensions, polynomial roots, and group theory concepts.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLzUeAPxtWcqzr80lS25FrzMn7a36BuXhj

**Tags:** abstract-algebra, group-theory, ring-theory, field-theory, galois-theory

## Books

### 1. Galois Theory

**Author:** Ian Stewart

A widely used undergraduate textbook that develops Galois theory from its historical motivation, the solution of polynomial equations by radicals. It covers field extensions, ruler-and-compass constructions, the Galois correspondence and insolubility of the quintic, preparing readers to prove these classical results themselves.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0412108003?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, abstract-algebra

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