Regulator of a number field (reviewed)

Let $\sigma_1,\ldots,\sigma_{r_1}$ be the real embeddings of a number field $K$ into the complex numbers $\mathbb{C}$, and $\sigma_{r_1+ 1},\ldots,\sigma_{r_1+r_2}$ be complex embeddings of $K$ into $\C$ such that no two are complex conjugate. Let $u_1,\ldots,u_r$ be a set of fundamental units of $K$. Then $r = r_1 + r_2 -1$.

Let $M$ be the $(r_1+ r_2-1)\times (r_1+r_2)$ matrix $(d_i\log{ \sigma_j(u_i)})$, where $d_i=1$ if $i\leq r_1$, i.e, if $\sigma_i$ is a real embedding, and $d_i=2$ otherwise, i.e., if $\sigma_i$ is a complex embedding. The sum of the columns of $M$ is the zero vector.

The regulator of $K$ is the absolute value of the determinant of the sub-matrix of $M$ where one column is removed. Its value is independent of the choice of column which is removed.