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[parent] Minkowski's constant (Corollary)

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:
$\displaystyle \vert{\bf N}(\mathfrak{A})\vert \leq M_K \sqrt{\vert D_K\vert}$
where $ {\bf N}(\mathfrak{A})$ denotes the absolute norm of $ \mathfrak{A}$ and
$\displaystyle M_K=\frac{n!}{n^n} \left(\frac{4}{\pi}\right)^{r_2}.$
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. $ \vert D_K\vert>1$. In particular, there is at least one rational prime $ p\in \mathbb{Z}$ which ramifies in $ K$.

See the entry on discriminants for the relationship between $ D_K$ and the ramification of primes.



"Minkowski's constant" is owned by alozano.
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See Also: ideal class, Stirling's approximation, discriminant, topics on ideal class groups and discriminants

Also defines:  Minkowski's theorem on ideal classes
Keywords:  ideal class group, discriminant

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Attachments:
Hermite's theorem (Corollary) by alozano
using Minkowski's constant to find a class number (Example) by alozano
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Cross-references: primes, ramifies, rational prime, absolute value, approximations, Stirling's formula, absolute norm, ideal, ideal class, class group, real and complex embeddings, number, degree, discriminant, number field, convex regions, Minkowski's theorem
There are 4 references to this entry.

This is version 1 of Minkowski's constant, born on 2005-02-24.
Object id is 6819, canonical name is MinkowskisConstant.
Accessed 2983 times total.

Classification:
AMS MSC: 11H06 (Number theory :: Geometry of numbers :: Lattices and convex bodies)
 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)
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