Minkowski's constantMinkowskisConstant2005-02-24 08:24:382005-02-24 08:24:38CorollaryMinkowski's theoremMinkowski's theorem on ideal classesideal class groupdiscriminant% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}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.
\begin{thm}[Minkowski's Theorem]
\label{thm1}
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 $\Rats$, where $r_1$ and $r_2$ are the number of real and complex embeddings, respectively. The class group of $K$ is denoted by $\Cl(K)$. In any ideal class $C\in \Cl(K)$, 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
$$M_K=\frac{n!}{n^n} \left(\frac{4}{\pi}\right)^{r_2}.$$
\end{thm}
\begin{defn}
The constant $M_K$, as in the theorem, is usually called the Minkowski's constant.
\end{defn}
In the applications, one uses Stirling's formula to find approximations of Minkowski's constant. The following is an immediate corollary of Theorem \ref{thm1}.
\begin{cor}
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 $p\in \Ints$ which ramifies in $K$.
\end{cor}
See the entry \PMlinkname{on discriminants}{DiscriminantOfANumberField} for the relationship between $D_K$ and the ramification of primes.