Minkowski’s constant

The following is a corollary to the famous Minkowski’s theorem on lattices and convex regions. It was also found by Minkowski and sometimes also called Minkowski’s theorem.

Theorem 1 (Minkowski’s Theorem).

Let K be a number fieldMathworldPlanetmath and let DK be its discriminantPlanetmathPlanetmathPlanetmathPlanetmath. Let n=r1+2r2 be the degree of K over Q, where r1 and r2 are the number of real and complex embeddings, respectively. The class groupMathworldPlanetmath of K is denoted by Cl(K). In any ideal class CCl(K), there exists an ideal AC such that:


where N(A) denotes the absolute norm of A and

Definition 1.

The constant MK, as in the theorem, is usually called the Minkowski’s constant.

In the applications, one uses Stirling’s formula to find approximations of Minkowski’s constant. The following is an immediate corollary of Theorem 1.

Corollary 1.

Let K be an arbitrary number field. Then the absolute valueMathworldPlanetmathPlanetmathPlanetmath of the discriminant of K, DK, is greater than 1, i.e. |DK|>1. In particular, there is at least one rational prime pZ which ramifies in K.

See the entry on discriminants (http://planetmath.org/DiscriminantOfANumberField) for the relationship between DK and the ramification of primes.

Title Minkowski’s constant
Canonical name MinkowskisConstant
Date of creation 2013-03-22 15:05:33
Last modified on 2013-03-22 15:05:33
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Corollary
Classification msc 11H06
Classification msc 11R29
Related topic IdealClass
Related topic StirlingsApproximation
Related topic DiscriminantOfANumberField
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups
Related topic ProofOfMinkowskisBound
Defines Minkowski’s theorem on ideal classes