---
title: Number Fields
description: This area of algebraic number theory studies finite-dimensional field extensions of the rational numbers. Learners will understand the ring of integers, prime ideal factorization, class groups, and how to generalize arithmetic concepts beyond standard integers.
category: mathematics
subcategory: number-theory
difficulty: beginner, intermediate, advanced
url: /subject/number-fields
---

# Number Fields

This area of algebraic number theory studies finite-dimensional field extensions of the rational numbers. Learners will understand the ring of integers, prime ideal factorization, class groups, and how to generalize arithmetic concepts beyond standard integers.

## Available Resources

2 Books • 2 Courses • 2 Websites

## Courses

### 1. Algebraic Number Theory

Graduate lecture series from IISc Bengaluru, taught by Mahesh Kakde, on number fields: rings of integers, ideal factorization in Dedekind domains, discriminants and ramification, the class group, and the unit theorem. Assumes a solid abstract algebra background.

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

**Link:** https://www.youtube.com/playlist?list=PLgMDNELGJ1CZQflgZinHm-rSNmYoAPtQb

**Tags:** algebraic-number-theory, number-fields, dedekind-domains, ideal-class-group, ramification

### 2. Number Theory I (MIT 18.785)

**Author:** Andrew Sutherland

Graduate algebraic number theory: Dedekind domains, ideal factorization, completions and local fields, ramification, the different and discriminant, class groups and Dirichlet's unit theorem. Complete lecture notes that read as a textbook come with problem sets.

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

**Link:** https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2019/

**Tags:** algebraic-number-theory, number-fields, dedekind-domains, local-fields, class-groups

## Books

### 1. Algebraic Number Theory

**Author:** Jürgen Neukirch

Graduate text building algebraic number theory from rings of integers and ideal class groups through valuations, local fields, Riemann–Roch for number fields, and zeta and L-functions. Rigorous and dense; requires strong commutative algebra and Galois theory.

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

**Link:** https://www.amazon.com/dp/3540653996?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, number-theory

### 2. A Course in Computational Algebraic Number Theory

**Author:** Henri Cohen

Graduate reference on algorithms for computing in number fields: lattice reduction, polynomial factorization, rings of integers, class groups and units, and elliptic curves. Pairs each algorithm with the theory behind it; many underlie the PARI/GP system.

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

**Link:** https://www.amazon.com/dp/3540556400?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, number-theory

## Websites

### 1. numbertheory.org

Numbertheory.org is a comprehensive online hub for learning number theory, offering curated tutorials, lecture notes, bibliographies, and a wide collection of links to papers, software, and related resources.

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

**Link:** https://numbertheory.org

**Tags:** websites, mathematics-statistics, mathematics

### 2. 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

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