# 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 $n\mathrm{=}{r}_{\mathrm{1}}\mathrm{+}\mathrm{2}\mathit{}{r}_{\mathrm{2}}$ be the degree of $K$ over $\mathrm{Q}$, where ${r}_{\mathrm{1}}$ and ${r}_{\mathrm{2}}$ are the number of real and complex embeddings, respectively. The class group^{} of $K$ is denoted by $\mathrm{Cl}\mathit{}\mathrm{(}K\mathrm{)}$. In any ideal class $C\mathrm{\in}\mathrm{Cl}\mathit{}\mathrm{(}K\mathrm{)}$, there exists an ideal $\mathrm{A}\mathrm{\in}C$ such that:

$$|\mathbf{N}(\U0001d504)|\le {M}_{K}\sqrt{|{D}_{K}|}$$ |

where $\mathrm{N}\mathit{}\mathrm{(}\mathrm{A}\mathrm{)}$ denotes the absolute norm of $\mathrm{A}$ 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 $\mathrm{1}$, i.e. $\mathrm{|}{D}_{K}\mathrm{|}\mathrm{>}\mathrm{1}$. In particular, there is at least one rational prime $p\mathrm{\in}\mathrm{Z}$ 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 |