Algebraic number fields

Introduction

An **algebraic number field** is a finite degree (finite-dimensional) field extension of the field of rational numbers, denoted by \(\mathbb{Q}\). Algebraic number fields are central objects of study in algebraic number theory, a branch of number theory that uses techniques from abstract algebra to study the properties of numbers.

Basic Definitions

Field Extension

A **field extension** \(K/\mathbb{Q}\) is a pair of fields such that \(\mathbb{Q} \subseteq K\). The field \(K\) is called an extension field of \(\mathbb{Q}\), and \(\mathbb{Q}\) is called the base field. The degree of the extension, denoted \([K : \mathbb{Q}]\), is the dimension of \(K\) as a vector space over \(\mathbb{Q}\).

Algebraic Numbers

An **algebraic number** is a complex number that is a root of a non-zero polynomial with rational coefficients. For example, \(\sqrt{2}\) is an algebraic number because it is a root of the polynomial \(x^2 - 2 = 0\).

Algebraic Number Field

An **algebraic number field** is a field extension \(K/\mathbb{Q}\) such that every element of \(K\) is an algebraic number. In other words, every element of \(K\) satisfies a polynomial equation with coefficients in \(\mathbb{Q}\).

Examples of Algebraic Number Fields

Quadratic Fields

A **quadratic field** is an algebraic number field of degree 2 over \(\mathbb{Q}\). It can be written in the form \(\mathbb{Q}(\sqrt{d})\), where \(d\) is a square-free integer. For example, \(\mathbb{Q}(\sqrt{-1})\) is the field of complex numbers with rational real and imaginary parts.

Cyclotomic Fields

A **cyclotomic field** is an algebraic number field obtained by adjoining a primitive root of unity to \(\mathbb{Q}\). For example, the field \(\mathbb{Q}(\zeta_n)\), where \(\zeta_n\) is a primitive \(n\)-th root of unity, is a cyclotomic field.

Number Fields of Higher Degree

Fields of higher degree can be constructed by adjoining more complex algebraic numbers. For instance, the field \(\mathbb{Q}(\sqrt[3]{2})\) is a cubic field, which is a degree 3 extension of \(\mathbb{Q}\).

Structure of Algebraic Number Fields

Ring of Integers

The **ring of integers** of an algebraic number field \(K\), denoted \(\mathcal{O}_K\), is the set of all elements in \(K\) that are roots of monic polynomials with coefficients in \(\mathbb{Z}\). For example, in the quadratic field \(\mathbb{Q}(\sqrt{d})\), the ring of integers is \(\mathbb{Z}[\sqrt{d}]\).

Units and Class Group

The **units** of \(\mathcal{O}_K\) are the elements that have a multiplicative inverse in \(\mathcal{O}_K\). The **class group** of \(K\) measures the failure of unique factorization in \(\mathcal{O}_K\). It is defined as the quotient group of the group of fractional ideals by the subgroup of principal ideals.

Galois Group

The **Galois group** of a field extension \(K/\mathbb{Q}\) is the group of field automorphisms of \(K\) that fix \(\mathbb{Q}\). If \(K/\mathbb{Q}\) is a Galois extension, then the Galois group is a finite group whose order is equal to the degree of the extension.

Applications and Importance

Algebraic number fields play a crucial role in various areas of mathematics, including the proof of Fermat's Last Theorem, the development of class field theory, and the study of Diophantine equations. They also have applications in cryptography, particularly in the construction of certain types of public-key cryptosystems.