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# number field

###### Definition 1.

A field which is a finite extension of $\mathbb{Q}$, the rational numbers, is called a number field (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 $\{\zeta_{n}^{a}:\gcd(a,n)=1,\ 1\leq a<n\}$.

###### 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:

$F^{+}=\mathbb{Q}(\zeta_{p}+\zeta_{p}^{{-1}}).$ |

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## Corrections

capitalization by Mathprof ✓

shorter name by pahio ✓

d \neq 1 by Wkbj79 ✓

Defines by pahio ✓