class number formulaClassNumberFormula2003-08-29 13:37:122003-08-29 13:53:45Theoremclass number formulaclass numberDedekind zeta function% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Nats}{\mathbb{N}}
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\newcommand{\Rats}{\mathbb{Q}}Let $K$ be a number field with $[K:\Rats]=n=r_1+2r_2$, where $r_1$
denotes the number of real embeddings of $K$, and $2r_2$ is the
number of complex embeddings of $K$. Let
$$\zeta_K(s)$$
be the Dedekind zeta function of $K$. Also define the following
invariants:
\begin{enumerate}
\item $h_K$ is the class number, the number of elements in the
ideal class group of $K$.
\item $\operatorname{Reg}_K$ is the regulator of $K$.
\item $\omega_K$ is the number of roots of unity contained in $K$.
\item $D_K$ is the discriminant of the extension $K/\Rats$.
\end{enumerate}
Then:
\begin{thm}[Class Number Formula]
The Dedekind zeta function of $K$, $\zeta_K(s)$ converges
absolutely for $\Re(s)>1$ and extends to a meromorphic function
defined for $\Re(s)>1-\frac{1}{n}$ with only one simple pole at
$s=1$. Moreover:
$$\lim_{s\to 1}
(s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot
\operatorname{Reg}_K}{\omega_K \cdot \sqrt{\mid D_K \mid}}$$
\end{thm}
Note: This is the most general ``class number formula''. In
particular cases, for example when $K$ is a cyclotomic extension
of $\Rats$, there are particular and more refined class number formulas.