Definition 1.

Let K be a number fieldMathworldPlanetmath, let dK be its discriminantPlanetmathPlanetmathPlanetmathPlanetmath and let n=[K:Q] be the degree over Q. The quantity:


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 fieldMathworldPlanetmath of a number field has the same root-discriminant as the number field.


Notice that the relative discriminant ideal (or different) for E/F is the ring of integersMathworldPlanetmath in F. Therefore we have:


The results follows by taking [E:]-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