# the ring of integers of a number field is finitely generated over $\mathbb{Z}$

###### Theorem.

Let $K$ be a number field of degree $n$ over $\mathrm{Q}$ and let ${\mathrm{O}}_{K}$ be the ring of integers^{} of $K$. The ring ${\mathrm{O}}_{K}$ is a free abelian group^{} of rank $n$. In other words, there exists a finite integral basis (with $n$ elements) for $K$, i.e. there exist algebraic integers^{} ${\alpha}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{\alpha}_{n}$ such that every element of ${\mathrm{O}}_{K}$ can be expressed uniquely as a $\mathrm{Z}$-linear combination^{} of the ${\alpha}_{i}$.

###### Corollary.

Every ideal of ${\mathrm{O}}_{K}$ is finitely generated^{}.

###### Proof of the corollary.

This is the first step to prove that ${\mathcal{O}}_{K}$ is a Dedekind domain^{}. Notice that the field of fractions^{} of ${\mathcal{O}}_{K}$ is the field $K$ itself. Therefore, by definition, ${\mathcal{O}}_{K}$ is integrally closed^{} in $K$.

Title | the ring of integers of a number field is finitely generated over $\mathbb{Z}$ |
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Canonical name | TheRingOfIntegersOfANumberFieldIsFinitelyGeneratedOvermathbbZ |

Date of creation | 2013-03-22 15:08:22 |

Last modified on | 2013-03-22 15:08:22 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 7 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 13B22 |