regulator

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

$${\sigma }_i:K\hookrightarrow ℝ,1\le i\le r_1$$

while $r_2$ is half of the number of complex embeddings:

$${\tau }_j:K\hookrightarrow ℂ,1\le j\le r_2$$

Note that $\{{\tau }_j,{\overline{\tau }}_j\mid 1\le j\le r_2\}$ areall the complex embeddings of $K$. Let $r=r_1+r_2$ and for$1\le i\le r$ define the “norm” in $K$ corresponding to eachembedding:

$$\parallel \cdot {\parallel }_i:K^{\times }\to {ℝ}^{+}$$

$${\parallel \alpha \parallel }_i=\mid {\sigma }_i(\alpha )\mid ,1\le i\le r_1$$

$${\parallel \alpha \parallel }_{r_1+j}={\mid {\tau }_j(\alpha )\mid }^2,1\le j\le r_2$$

Let ${𝒪}_K$ be the ring of integers of$K$. By Dirichlet’s unit theorem, we know that the rank of theunit group ${𝒪}_K^{\times }$ is exactly $r-1=r_1+r_2-1$.Let

$$\{{ϵ}_1,{ϵ}_2,\mathrm{\dots },{ϵ}_{r-1}\}$$

be a fundamental system of generators of ${𝒪}_K^{\times }$modulo roots of unity (this is, modulo the torsion subgroup). Let$A$ be the $r\times (r-1)$ matrix

and let $A_i$ be the $(r-1)\times (r-1)$ matrix obtainedby deleting the $i$-th row from $A$, $1\le i\le r$. It can bechecked that the determinant of $A_i$, $det{A}_i$, is independentup to sign of the choice of fundamental system of generators of${𝒪}_K^{\times }$ and is also independent of the choice of$i$.

Definition.

The regulator of $K$ is defined to be

$${\mathrm{Reg}}_K=\mid det{A}_1\mid$$

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