number field

Definition 1.

A field which is a finite extension of $\mathrm{Q}$, the rational numbers, is called a number field (sometimes called algebraic number field). If the degree of the extension $K\mathrm{/}\mathrm{Q}$ is $n$ then we say that $K$ is a number field of degree $n$ (over $\mathrm{Q}$).

Example 1.

The field of rational numbers $\mathbb{Q}$ is a number field.

Example 2.

Let $K=\mathbb{Q}(\sqrt{d})$, where $d\ne 1$ is a square-free non-zero integer and $\sqrt{d}$ stands for any of the roots of $x^2-d=0$ (note that if $\sqrt{d}\in K$ then $-\sqrt{d}\in K$ as well). Then $K$ is a number field and $[K:\mathbb{Q}]=2$. We can explictly describe all elements of $K$ as follows:

$$K=\{t+s\sqrt{d}:t,s\in \mathbb{Q}\}.$$

Definition 2.

A number field $K$ such that the degree of the extension $K\mathrm{/}\mathrm{Q}$ is $\mathrm{2}$ is called a quadratic number field.

In fact, if $K$ is a quadratic number field, then it is easy to show that $K$ is one of the fields described in Example $2$.

Example 3.

Let $K_n=\mathbb{Q}({\zeta }_n)$ be a cyclotomic extension of $\mathbb{Q}$, where ${\zeta }_n$ is a primitive $n$th root of unity. Then $K$ is a number field and

$$[K:\mathbb{Q}]=\phi (n)$$

where $\phi (n)$ is the Euler phi function. In particular, $\phi (3)=2$, therefore $K_3$ is a quadratic number field (in fact $K_3=\mathbb{Q}(\sqrt{-3})$). We can explicitly describe all elements of $K$ as follows:

$$K_n=\{q_0+q_1{\zeta }_n+q_2{\zeta }_n^2+\mathrm{\dots }+q_{n-1}{\zeta }_n^{n-1}:q_i\in \mathbb{Q}\}.$$

In fact, one can do better. Every element of $K_n$ can be uniquely expressed as a rational combination of the $\phi (n)$ elements .

Example 4.

Let $K$ be a number field. Then any subfield $L$ with $\mathbb{Q}\subseteq L\subseteq K$ is also a number field. For example, let $p$ be a prime number and let $F=\mathbb{Q}({\zeta }_p)$ be a cyclotomic extension of $\mathbb{Q}$, where ${\zeta }_p$ is a primitive $p$th root of unity. Let $F^{+}$ be the maximal real subfield of $F$. $F^{+}$ is a number field and it can be shown that:

$$F^{+}=\mathbb{Q}({\zeta }_p+{\zeta }_p^{-1}).$$

Title	number field
Canonical name	NumberField
Date of creation	2013-03-22 12:04:09
Last modified on	2013-03-22 12:04:09
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	17
Author	alozano (2414)
Entry type	Definition
Classification	msc 11-00
Synonym	algebraic number field
Related topic	AlgebraicNumberTheory
Related topic	ExamplesOfPrimeIdealDecompositionInNumberFields
Related topic	ExamplesOfFields
Related topic	AbelianExtensionsOfQuadraticImaginaryNumberFields
Related topic	NumberTheory
Related topic	ResidueDegree
Related topic	Regulator
Related topic	DiscriminantIdeal
Related topic	ClassNumber2
Related topic	ExistenceOfHilbertClassField
Related topic	Multiplicat
Defines	quadratic number field
Defines	quadratic field