---
title: Fields
description: Fields are algebraic structures where addition, subtraction, multiplication, and division are defined and behave like rational numbers. Learners will understand field extensions, finite fields, and how these systems apply to cryptography, coding theory, and polynomial equations.
category: mathematics
subcategory: abstract-algebra
difficulty: beginner, intermediate, advanced
url: /subject/fields
---

# Fields

Fields are algebraic structures where addition, subtraction, multiplication, and division are defined and behave like rational numbers. Learners will understand field extensions, finite fields, and how these systems apply to cryptography, coding theory, and polynomial equations.

## Available Resources

1 Courses • 3 Websites

## Websites

### 1. Stacks Project

Open, collaboratively written reference on algebraic geometry whose commutative algebra chapter is cited by tag number. This tag is a lemma on Jacobson rings, showing that localizing at one element preserves the Jacobson property. Readers need solid ring theory to follow it.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://stacks.math.columbia.edu/tag/00G6

**Tags:** commutative algebra, jacobson rings, localization, algebraic geometry

### 2. nLab: Field

Research-level wiki entry defining fields as commutative rings with invertible nonzero elements, then examining constructive variants, the category of fields and links to related structures. Readers gain the categorical and constructive viewpoint that standard algebra textbooks usually omit.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/field

**Tags:** field theory, category theory, constructive mathematics, commutative rings

### 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. Harvard MATH E-222: Abstract Algebra

**Author:** Benedict Gross

Recorded Harvard Extension School lectures by Benedict Gross, following Michael Artin's textbook Algebra. Covers groups and group actions, vector spaces and linear operators, rings, and fields, with proofs throughout. Viewers gain a rigorous first grounding in algebraic structures and symmetry.

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

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

**Tags:** abstract algebra, group theory, group actions, linear algebra, rings and fields

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