# root-discriminant

###### Definition 1.

Let $K$ be a number field^{}, let ${d}_{K}$ be its discriminant^{} and let $n\mathrm{=}\mathrm{[}K\mathrm{:}\mathrm{Q}\mathrm{]}$ be the degree over $\mathrm{Q}$. The quantity:

$$|\sqrt[n]{{d}_{K}}|$$ |

is called the root-discriminant of $K$ and it is usually denoted by ${\mathrm{rd}}_{K}$.

The following lemma is one of the motivations for the previous definition:

###### Lemma 1.

Let $E\mathrm{/}F$ be an extension of number fields which is unramified at all finite primes. Then ${\mathrm{rd}}_{E}\mathrm{=}{\mathrm{rd}}_{F}$. In particular, the Hilbert class field^{} of a number field has the same root-discriminant as the number field.

###### Proof.

Notice that the relative discriminant ideal (or different) for $E/F$ is the ring of integers^{} in $F$. Therefore we have:

$$|{d}_{E}|={|{d}_{F}|}^{[E:F]}$$ |

The results follows by taking $[E:\mathbb{Q}]$-th roots on both sides of the previous equation. ∎

Title | root-discriminant |
---|---|

Canonical name | Rootdiscriminant |

Date of creation | 2013-03-22 15:05:44 |

Last modified on | 2013-03-22 15:05:44 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 11R29 |

Synonym | root discriminant |

Related topic | ExistenceOfHilbertClassField |