Regulator of an algebraic number field
The number 
that is, by definition, equal to 1 if 
is the field 
or an imaginary quadratic extension of 
, and to 
in all other cases, where 
is the rank of the group 
of units of the field 
(see Algebraic number; Algebraic number theory) and 
is the 
-dimensional volume of the basic parallelepipedon of the 
-dimensional lattice in 
that is the image of 
under its logarithmic mapping 
into 
. The homomorphism 
is defined as follows: Let 
be all real and let 
be all pairwise complex non-conjugate isomorphisms of 
into 
; 
. Then 
(see Dirichlet theorem on units), and 
is defined by the formula
 |
where
 |
The image of 
under 
is an 
-dimensional lattice in 
lying in the plane 
(where the 
are the canonical coordinates).
Units 
for which 
form a basis of the lattice 
are known as fundamental units of 
, and
 |
There are other formulas linking the regulator with other invariants of the field 
(see, for example, Discriminant, 3).
If instead of 
one considers the intersection of this group with an order 
of 
, then the regulator 
of 
can be defined in the same way.
References
| [1] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966) |
| [2] | S. Lang, "Algebraic number theory" , Addison-Wesley (1970) |
Regulator of an algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regulator_of_an_algebraic_number_field&oldid=17455