regulatorRegulator2003-08-29 13:30:112006-11-09 08:31:40Definitionregulator of a number fieldregulatorunit% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}Let $K$ be a number field with $[K:\Rats]=n=r_1+2r_2$. Here $r_1$
denotes the number of real embeddings:
$$\sigma_i\colon K \hookrightarrow \Reals,\quad 1\leq i\leq r_1$$
while $r_2$ is half of the number of complex embeddings:
$$\tau_j\colon K \hookrightarrow \Complex,\quad 1\leq j\leq r_2$$
Note that $\{\tau_j, \bar{\tau}_j\mid 1\leq j\leq r_2\}$ are
\emph{all} the complex embeddings of $K$. Let $r=r_1+r_2$ and for
$1\leq i\leq r$ define the ``norm'' in $K$ corresponding to each
embedding:
$$ \parallel \cdot \parallel _i\colon K^{\times} \to \Reals^+$$
$$ \parallel \alpha \parallel_i = \mid\sigma_i(\alpha)\mid, \quad
1\leq i \leq r_1$$
$$ \parallel \alpha \parallel_{r_1+j} = \mid\tau_j(\alpha)\mid^2, \quad
1\leq j \leq r_2$$ Let $\mathcal{O}_K$ be the ring of integers of
$K$. By Dirichlet's unit theorem, we know that the rank of the
unit group $\mathcal{O}_K^{\times}$ is exactly $r-1=r_1+r_2-1$.
Let
$$\{ \epsilon_1,\epsilon_2,\ldots,\epsilon_{r-1}\}$$
be a fundamental system of generators of $\mathcal{O}_K^{\times}$
modulo roots of unity (this is, modulo the torsion subgroup). Let
$A$ be the $r\times (r-1)$ matrix $$A=\left(
\begin{array}{cccc}
\log \parallel \epsilon_1 \parallel_1 & \log \parallel \epsilon_2 \parallel_1 & \ldots & \log \parallel \epsilon_{r-1} \parallel_1 \\
\log \parallel \epsilon_1 \parallel_2 & \log \parallel \epsilon_2 \parallel_2 & \ldots & \log \parallel \epsilon_{r-1} \parallel_2 \\
\vdots & \vdots & \ddots & \vdots \\
\log \parallel \epsilon_1 \parallel_r & \log \parallel \epsilon_2 \parallel_r & \ldots & \log \parallel \epsilon_{r-1} \parallel_r \\
\end{array}
\right)$$ and let $A_i$ be the $(r-1)\times(r-1)$ matrix obtained
by deleting the $i$-th row from $A$, $1\leq i\leq r$. It can be
checked that the determinant of $A_i$, $\det{A_i}$, is independent
up to sign of the choice of fundamental system of generators of
$\mathcal{O}_K^{\times}$ and is also independent of the choice of
$i$.
\begin{defn}
The \emph{regulator of} $K$ is defined to be
$$\operatorname{Reg}_K=\mid\det{A_1}\mid$$
\end{defn}
The regulator is one of the main ingredients in the analytic class number formula for number fields.
\begin{thebibliography}{9}
\bibitem{marcus} Daniel A. Marcus, {\em Number Fields},
Springer, New York.
\bibitem{lang} Serge Lang, {\em Algebraic Number Theory}. Springer-Verlag, New York.
\end{thebibliography}