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 field and let $D_{K}$ be its discriminant. Let $n=r_{1}+2r_{2}$ be the degree of $K$ over $\mathbb{Q}$ , where $r_{1}$ and $r_{2}$ are the number of real and complex embeddings, respectively. The class group of $K$ is denoted by $\operatorname{Cl}(K)$ . In any ideal class $C\in\operatorname{Cl}(K)$ , there exists an ideal $\mathfrak{A}\in C$ such that:

$|{\bf N}(\mathfrak{A})|\leq M_{K}\sqrt{|D_{K}|}$

where ${\bf N}(\mathfrak{A})$ denotes the absolute norm of $\mathfrak{A}$ and

$M_{K}=\frac{n!}{n^{n}}\left(\frac{4}{\pi}\right)^{r_{2}}.$

Definition 1.

The constant $M_{K}$ , 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 value of the discriminant of $K$ , $D_{K}$ , is greater than $1$ , i.e. $|D_{K}|>1$ . In particular, there is at least one rational prime $p\in\mathbb{Z}$ which ramifies in $K$ .

See the entry on discriminants (http://planetmath.org/DiscriminantOfANumberField) for the relationship between $D_{K}$ 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