---
title: Algebraic Numbers
description: This field studies complex numbers that are roots of non-zero polynomials with rational coefficients. Learners will understand the algebraic properties of these numbers, analyze field extensions, and distinguish them from transcendental numbers.
category: mathematics
subcategory: number-theory
difficulty: beginner, intermediate, advanced
url: /subject/algebraic-numbers
---

# Algebraic Numbers

This field studies complex numbers that are roots of non-zero polynomials with rational coefficients. Learners will understand the algebraic properties of these numbers, analyze field extensions, and distinguish them from transcendental numbers.

## Available Resources

2 Books • 1 Courses • 4 Websites

## Websites

### 1. nLab: Algebraic Number

Concise nLab wiki entry defining algebraic numbers as roots of polynomials with rational coefficients, with links to algebraic integers, number fields and algebraic closure. Terse and category-theoretic in style, so it suits readers who already know abstract algebra.

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

**Link:** https://ncatlab.org/nlab/show/algebraic+number

**Tags:** algebraic-numbers, algebraic-integers, number-fields, field-theory, reference

### 2. Keith Conrad's Expository Papers

**Author:** Keith Conrad

Collection of free short expository handouts by Keith Conrad. The fields/Galois theory and algebraic number theory sections work through factoring in quadratic fields, ideal factorization, discriminants, class group calculations, and Dirichlet's unit theorem with explicit computed examples; includes a transcendence-of-e proof. Fills the gap the textbooks leave: fully worked concrete examples (rings of integers without a power basis, Minkowski-bound class group computations, trace and norm).

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

**Link:** https://kconrad.math.uconn.edu/blurbs/

### 3. Algebraic Number Theory (Course Notes)

**Author:** J.S. Milne

J.S. Milne's free 166-page graduate course notes (v3.08, 2020) covering rings of integers, Dedekind domains and ideal factorization, finiteness of the class number, the unit theorem, cyclotomic extensions, valuations and local fields. Readers finish able to compute in number fields.

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

**Link:** https://www.jmilne.org/math/CourseNotes/ant.html

### 4. 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. Algebraic Number Theory

Learn algebraic number theory with Prof. Benedict Gross! Explore algebraic numbers and their properties in this comprehensive course.

**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

## Books

### 1. Numbers: Rational and Irrational

**Author:** Ivan Niven

Ivan Niven's 140-page New Mathematical Library volume building from integers and rationals to irrational, algebraic and transcendental numbers using only school algebra. Readers can prove specific numbers irrational, understand rational approximation, and follow a proof that transcendental numbers exist.

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

**Link:** https://www.amazon.com/dp/0883856018?tag=edmonddante07-20

### 2. Number Fields (2nd Edition)

**Author:** Daniel A. Marcus

The one book to work through for self-study: lower prerequisites than Milne, avoids local methods, teaches through exercises. Daniel Marcus's exercise-driven first text on algebraic numbers, requiring only undergraduate abstract algebra. Covers number fields and rings of integers, prime decomposition, Galois theory applied to primes, the ideal class group, units, and the Dedekind zeta function, with several hundred exercises.

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

**Link:** https://www.amazon.com/dp/3319902326?tag=edmonddante07-20

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