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 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

The regulator is one of the main ingredients in the analytic class number formula for number fields.

Bibliography

1

Daniel A. Marcus, Number Fields, Springer, New York.

2

Serge Lang, Algebraic Number Theory. Springer-Verlag, New York.