# 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