Regulator (number theory)

In number theory , the regulator describes a quantity that provides information about the units of an algebraic number field . Such a regulator is assigned to each number body . ${\displaystyle K}$ ${\displaystyle R_{K}}$

definition

Let be an algebraic number field with degree of expansion . Let be the number of real embeddings of and the number of complex embeddings, so it holds . Then the free part of the unit group of the whole closure from in is isomorphic to according to Dirichlet's unit set . If one forms the unit group over ${\displaystyle K}$ ${\displaystyle [K:\mathbb {Q} ]=n}$ ${\displaystyle r}$ ${\displaystyle K}$ ${\displaystyle 2s}$ ${\displaystyle n=r+2s}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle K}$ ${\displaystyle \mathbb {Z} ^{r+s-1}}$ ${\displaystyle {\mathcal {O}}_{K}^{\times }}$

${\displaystyle a\mapsto (\ln(|\rho _{1}(a)|),\dots ,\ln(|\rho _{r}(a)|),2\ln(|\sigma _{1}(a)|),\dots ,2\ln(|\sigma _{s}(a)|))}$

in the ab, where the real embeddings and the complex embeddings are, then the image is a -dimensional lattice of the volume . The regulator is now defined as ${\displaystyle \mathbb {R} ^{r+s}}$ ${\displaystyle \rho _{1},\dots ,\rho _{r}}$ ${\displaystyle \sigma _{1},{\overline {\sigma _{1}}}\dots ,\sigma _{s},{\overline {\sigma _{s}}}}$ ${\displaystyle (r+s-1)}$ ${\displaystyle V}$ ${\displaystyle R_{K}}$

${\displaystyle R_{K}:={\frac {V}{\sqrt {r+s}}}.}$

It represents an important quantity of the number field and appears again , for example, in the class number formula .

Generalizations

Generalizations of this term include the Borel regulator and the Beilinson regulator . To distinguish it from these, the special case described in this article is also referred to as the Dirichletscher regulator .

literature

• SI Borevich, IR Shafarevich: Number Theory . Birkhäuser Verlag, 1966.