# 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 |