PlanetMath: regulator

(more info)

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regulator

(Definition)

Let be a number field with $[K:\mathbb{Q}]=n=r_1+2r_2$ . Here denotes the number of real embeddings:

$\displaystyle \sigma_i\colon K \hookrightarrow \mathbb{R},\quad 1\leq i\leq r_1$

while

is half of the number of complex embeddings:

$\displaystyle \tau_j\colon K \hookrightarrow \mathbb{C},\quad 1\leq j\leq r_2$

Note that $\{\tau_j, \bar{\tau}_j\mid 1\leq j\leq r_2\}$ are all the complex embeddings of

. Let

and for $1\leq i\leq r$ define the “norm” in

corresponding to each embedding:

$\displaystyle \parallel \cdot \parallel _i\colon K^{\times} \to \mathbb{R}^+$

$\displaystyle \parallel \alpha \parallel_i = \mid\sigma_i(\alpha)\mid, \quad 1\leq i \leq r_1$

$\displaystyle \parallel \alpha \parallel_{r_1+j} = \mid\tau_j(\alpha)\mid^2, \quad 1\leq j \leq r_2$

Let $\mathcal{O}_K$ be the ring of integers of

. By Dirichlet's unit theorem, we know that the rank of the unit group $\mathcal{O}_K^{\times}$ is exactly

. Let

$\displaystyle \{ \epsilon_1,\epsilon_2,\ldots,\epsilon_{r-1}\}$

be a fundamental system of generators of $\mathcal{O}_K^{\times}$ modulo roots of unity (this is, modulo the torsion subgroup). Let

be the $r\times (r-1)$ matrix

$\begin{displaymath}A=\left( \begin{array}{cccc} \log \parallel \epsilon_1 \paral... ...log \parallel \epsilon_{r-1} \parallel_r \ \end{array}\right)\end{displaymath}$

and let

be the $(r-1)\times(r-1)$ matrix obtained by deleting the

-th row from

, $1\leq i\leq r$ . It can be checked that the determinant of

, $\det{A_i}$ , 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