# Regulator of an algebraic number field

$K$

The number $R _ {K}$ that is, by definition, equal to 1 if $K$ is the field $\mathbf Q$ or an imaginary quadratic extension of $\mathbf Q$, and to $v/ \sqrt {r+1}$ in all other cases, where $r$ is the rank of the group $E$ of units of the field $K$( see Algebraic number; Algebraic number theory) and $v$ is the $r$- dimensional volume of the basic parallelepipedon of the $r$- dimensional lattice in $\mathbf R ^ {r+1}$ that is the image of $E$ under its logarithmic mapping $l$ into $\mathbf R ^ {r+1}$. The homomorphism $l$ is defined as follows: Let $\sigma _ {1} \dots \sigma _ {s}$ be all real and let $\sigma _ {s+1} \dots \sigma _ {s+t}$ be all pairwise complex non-conjugate isomorphisms of $K$ into $\mathbf C$; $s + 2t = \mathop{\rm dim} _ {\mathbf Q} K$. Then ${r+1} = {s+t}$( see Dirichlet theorem on units), and $l: E \rightarrow \mathbf R ^ {r+1}$ is defined by the formula

$$l( \alpha ) = ( l _ {1} ( \alpha ) \dots l _ {s+t} ( \alpha )),$$

where

$$l _ {i} ( \alpha ) = \left \{ \begin{array}{ll} \mathop{\rm ln} | \sigma _ {i} ( \alpha ) | &\textrm{ if } 1 \leq i \leq s, \\ \mathop{\rm ln} | \sigma _ {i} ( \alpha ) | ^ {2} &\textrm{ if } {s+1} \leq i \leq {s+t}. \\ \end{array} \right .$$

The image of $E$ under $l$ is an $r$- dimensional lattice in $\mathbf R ^ {r+1}$ lying in the plane $\sum _ {i=} 0 ^ {r+1} x _ {i} = 0$( where the $x _ {i}$ are the canonical coordinates).

Units $\epsilon _ {1} \dots \epsilon _ {r}$ for which $l( e _ {1} ) \dots l( e _ {r} )$ form a basis of the lattice $l( E)$ are known as fundamental units of $K$, and

$$R _ {K} = \| \mathop{\rm det} ( l _ {i} ( \epsilon _ {j} )) _ {i,j= 1 } ^ {r} \| .$$

There are other formulas linking the regulator with other invariants of the field $K$( see, for example, Discriminant, 3).

If instead of $E$ one considers the intersection of this group with an order ${\mathcal O}$ of $K$, then the regulator $R _ {\mathcal O}$ of ${\mathcal O}$ can be defined in the same way.

How to Cite This Entry:
Regulator of an algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regulator_of_an_algebraic_number_field&oldid=49670
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article