root-discriminant
Definition 1.
Let K be a number field, let dK be its discriminant and let n=[K:Q] be the degree over Q. The quantity:
|n√dK| |
is called the root-discriminant of K and it is usually denoted by rdK.
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 rdE=rdF. 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:
|dE|=|dF|[E:F] |
The results follows by taking [E: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 |