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Difference between revisions of "Regulator of an algebraic number field"

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m (typos)
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is the field  $  \mathbf Q $
 
is the field  $  \mathbf Q $
 
or an imaginary quadratic extension of  $  \mathbf Q $,  
 
or an imaginary quadratic extension of  $  \mathbf Q $,  
and to  $  v/ \sqrt r+ 1 $
+
and to  $  v/ \sqrt {r+1} $
 
in all other cases, where  $  r $
 
in all other cases, where  $  r $
 
is the rank of the group  $  E $
 
is the rank of the group  $  E $
Line 24: Line 24:
 
is the  $  r $-
 
is the  $  r $-
 
dimensional volume of the basic parallelepipedon of the  $  r $-
 
dimensional volume of the basic parallelepipedon of the  $  r $-
dimensional lattice in  $  \mathbf R ^ {r+} 1 $
+
dimensional lattice in  $  \mathbf R ^ {r+1} $
 
that is the image of  $  E $
 
that is the image of  $  E $
 
under its logarithmic mapping  $  l $
 
under its logarithmic mapping  $  l $
into  $  \mathbf R ^ {r+} 1 $.  
+
into  $  \mathbf R ^ {r+1} $.  
 
The homomorphism  $  l $
 
The homomorphism  $  l $
 
is defined as follows: Let  $  \sigma _ {1} \dots \sigma _ {s} $
 
is defined as follows: Let  $  \sigma _ {1} \dots \sigma _ {s} $
be all real and let  $  \sigma _ {s+} 1 \dots \sigma _ {s+} t $
+
be all real and let  $  \sigma _ {s+1} \dots \sigma _ {s+t} $
 
be all pairwise complex non-conjugate isomorphisms of  $  K $
 
be all pairwise complex non-conjugate isomorphisms of  $  K $
 
into  $  \mathbf C $;  
 
into  $  \mathbf C $;  
 
$  s + 2t =  \mathop{\rm dim} _ {\mathbf Q}  K $.  
 
$  s + 2t =  \mathop{\rm dim} _ {\mathbf Q}  K $.  
Then  $  r+ 1 = s+ t $(
+
Then  $  {r+1} = {s+t} $(
see [[Dirichlet theorem|Dirichlet theorem]] on units), and  $  l:  E \rightarrow \mathbf R ^ {r+} 1 $
+
see [[Dirichlet theorem|Dirichlet theorem]] on units), and  $  l:  E \rightarrow \mathbf R ^ {r+1} $
 
is defined by the formula
 
is defined by the formula
  
 
$$  
 
$$  
l( \alpha )  =  ( l _ {1} ( \alpha ) \dots l _ {s+} t ( \alpha )),
+
l( \alpha )  =  ( l _ {1} ( \alpha ) \dots l _ {s+t} ( \alpha )),
 
$$
 
$$
  
Line 49: Line 49:
 
\begin{array}{ll}
 
\begin{array}{ll}
 
  \mathop{\rm ln}  | \sigma _ {i} ( \alpha ) |  &\textrm{ if }  1 \leq  i \leq  s,  \\
 
  \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.  \\
+
  \mathop{\rm ln}  | \sigma _ {i} ( \alpha ) |  ^ {2}  &\textrm{ if }  {s+1} \leq  i \leq  {s+t}.  \\
 
\end{array}
 
\end{array}
 
  \right .
 
  \right .
Line 57: Line 57:
 
under  $  l $
 
under  $  l $
 
is an  $  r $-
 
is an  $  r $-
dimensional lattice in  $  \mathbf R ^ {r+} 1 $
+
dimensional lattice in  $  \mathbf R ^ {r+1} $
lying in the plane  $  \sum _ {i=} 0 ^ {r+} 1 x _ {i} = 0 $(
+
lying in the plane  $  \sum _ {i=} 0 ^ {r+1} x _ {i} = 0 $(
 
where the  $  x _ {i} $
 
where the  $  x _ {i} $
 
are the canonical coordinates).
 
are the canonical coordinates).

Revision as of 21:46, 7 June 2020


$ 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.

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)
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=49558
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article