number field

Definition 1.

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

Example 1.

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

Example 2.

Let $K=\mathbb{Q}(\sqrt{d})$, where $d\neq 1$ 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 $K/\mathbb{Q}$ is $2$ 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

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

where $\varphi(n)$ is the Euler phi function. In particular, $\varphi(3)=2$, 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:

 $K_{n}=\{q_{0}+q_{1}\zeta_{n}+q_{2}\zeta_{n}^{2}+\ldots+q_{n-1}\zeta_{n}^{n-1}:% q_{i}\in\mathbb{Q}\}.$

In fact, one can do better. Every element of $K_{n}$ can be uniquely expressed as a rational combination of the $\varphi(n)$ elements $\{\zeta_{n}^{a}:\gcd(a,n)=1,\ 1\leq a.

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