# root-discriminant

###### Definition 1.

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

 $|\sqrt[n]{d_{K}}|$

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

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

###### Lemma 1.

Let $E/F$ be an extension of number fields which is unramified at all finite primes. Then $\operatorname{rd}_{E}=\operatorname{rd}_{F}$. 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 Rootdiscriminant 2013-03-22 15:05:44 2013-03-22 15:05:44 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 11R29 root discriminant ExistenceOfHilbertClassField