Number Fields

Fields Generated by Algebraic Numbers

A number field is a finite extension of the rational numbers. Concretely, it is a field $K$ satisfying

\mathbb{Q}\subseteq K\subseteq \mathbb{C}

and such that $K$ has finite dimension as a vector space over $\mathbb{Q}$ .

The simplest examples are obtained by adjoining an algebraic number $\alpha$ to $\mathbb{Q}$ . The resulting field is denoted

\mathbb{Q}(\alpha).

It is the smallest field containing both $\mathbb{Q}$ and $\alpha$ .

For example,

\mathbb{Q}(\sqrt2) = \{a+b\sqrt2:a,b\in\mathbb{Q}\}.

This field contains rational numbers and also the irrational number $\sqrt2$ , together with all sums, products, and quotients formed from them.

Degree of a Number Field

The degree of a number field $K$ over $\mathbb{Q}$ is the vector space dimension

[K:\mathbb{Q}].

K=\mathbb{Q}(\alpha),

then

[K:\mathbb{Q}] = \deg m_\alpha,

where $m_\alpha(x)$ is the minimal polynomial of $\alpha$ over $\mathbb{Q}$ .

For example, the minimal polynomial of $\sqrt2$ is

x^2-2,

[\mathbb{Q}(\sqrt2):\mathbb{Q}]=2.

The field $\mathbb{Q}(\sqrt[3]{2})$ has degree $3$ , because $\sqrt[3]{2}$ has minimal polynomial

x^3-2.

Quadratic Fields

A number field of degree $2$ is called a quadratic field. Every quadratic field has the form

\mathbb{Q}(\sqrt d),

where $d$ is a squarefree integer.

Examples include

\mathbb{Q}(\sqrt2), \qquad \mathbb{Q}(\sqrt5), \qquad \mathbb{Q}(i)=\mathbb{Q}(\sqrt{-1}).

Quadratic fields are the first nontrivial number fields and serve as a laboratory for algebraic number theory.

In a quadratic field, every element can be written uniquely as

a+b\sqrt d, \qquad a,b\in\mathbb{Q}.

The conjugate of this element is

a-b\sqrt d.

Arithmetic in Number Fields

Number fields allow arithmetic beyond the rational numbers while preserving algebraic structure.

\alpha,\beta\in K,

then

\alpha+\beta,\qquad \alpha-\beta,\qquad \alpha\beta,\qquad \frac{\alpha}{\beta}\quad(\beta\ne0)

also belong to $K$ .

This closure under field operations makes number fields suitable for studying equations whose natural solutions involve radicals or roots of polynomials.

For example, Pell equations lead naturally to

\mathbb{Q}(\sqrt D),

while higher reciprocity laws lead to cyclotomic fields such as

\mathbb{Q}(\zeta_n),

where $\zeta_n$ is a primitive $n$ -th root of unity.

Bases and Coordinates

Since $K$ is finite-dimensional over $\mathbb{Q}$ , every element of $K$ can be expressed in terms of a basis.

For

K=\mathbb{Q}(\alpha),

where $\alpha$ has degree $n$ , a standard basis is

1,\alpha,\alpha^2,\ldots,\alpha^{n-1}.

Thus every element of $K$ has a unique expression

c_0+c_1\alpha+\cdots+c_{n-1}\alpha^{n-1}, \qquad c_i\in\mathbb{Q}.

Higher powers of $\alpha$ can be reduced using the minimal polynomial.

For example, in $\mathbb{Q}(\sqrt[3]{2})$ , every element has the form

a+b\sqrt[3]{2}+c\sqrt[3]{4}, \qquad a,b,c\in\mathbb{Q}.

Embeddings

A number field can be embedded into the complex numbers in several ways.

An embedding is an injective field homomorphism

\sigma:K\to\mathbb{C}

that fixes every rational number.

K=\mathbb{Q}(\alpha),

then embeddings correspond to the conjugates of $\alpha$ , that is, the roots of its minimal polynomial.

For example, $\mathbb{Q}(\sqrt2)$ has two embeddings:

\sqrt2\mapsto \sqrt2,

and

\sqrt2\mapsto -\sqrt2.

The field $\mathbb{Q}(\sqrt[3]{2})$ has three complex embeddings, sending $\sqrt[3]{2}$ to the three roots of $x^3-2$ .

Ring of Integers

Inside a number field $K$ , one studies the algebraic integers contained in $K$ . These form a ring called the ring of integers of $K$ , denoted

\mathcal O_K.

This ring plays the role of $\mathbb{Z}$ inside $K$ .

For example,

\mathcal O_{\mathbb{Q}(i)}=\mathbb{Z}[i].

But in

\mathbb{Q}(\sqrt5),

the ring of integers is

\mathbb{Z}\left[\frac{1+\sqrt5}{2}\right],

not merely $\mathbb{Z}[\sqrt5]$ .

The ring of integers is the main object of arithmetic study in a number field.

Norm and Trace

Every finite extension $K/\mathbb{Q}$ has norm and trace maps

N_{K/\mathbb{Q}}:K\to\mathbb{Q},

\operatorname{Tr}_{K/\mathbb{Q}}:K\to\mathbb{Q}.

If the embeddings of $K$ into $\mathbb{C}$ are

\sigma_1,\ldots,\sigma_n,

then

N_{K/\mathbb{Q}}(\alpha) = \sigma_1(\alpha)\sigma_2(\alpha)\cdots\sigma_n(\alpha),

and

\operatorname{Tr}_{K/\mathbb{Q}}(\alpha) = \sigma_1(\alpha)+\sigma_2(\alpha)+\cdots+\sigma_n(\alpha).

In a quadratic field,

N(a+b\sqrt d)=a^2-db^2,

and

\operatorname{Tr}(a+b\sqrt d)=2a.

These maps generalize familiar arithmetic operations and are essential for studying divisibility and units.

Why Number Fields Matter

Number fields provide the natural setting for many problems in number theory.

They arise in:

Pell equations through quadratic fields,

sums of squares through Gaussian integers,

higher reciprocity through cyclotomic fields,

Fermat-type equations through algebraic factorizations,

and elliptic curves through fields generated by torsion points.

The central idea is that an equation over $\mathbb{Z}$ may become more transparent after enlarging the number system. In the larger field, hidden factorizations and symmetries appear.

Structural Role

Number fields extend the arithmetic of $\mathbb{Q}$ while remaining finite enough to be controlled.

They support notions of:

algebraic integer,
norm and trace,
ideals,
units,
ramification,
class group,
zeta function.

Thus they form the basic objects of algebraic number theory.

Many modern theories, including class field theory, modular forms, and the Langlands program, begin with the arithmetic of number fields.