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The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809502.png" /> that is, by definition, equal to 1 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809503.png" /> is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809504.png" /> or an imaginary quadratic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809505.png" />, and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809506.png" /> in all other cases, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809507.png" /> is the rank of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809508.png" /> of units of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809509.png" /> (see [[Algebraic number|Algebraic number]]; [[Algebraic number theory|Algebraic number theory]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095010.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095011.png" />-dimensional volume of the basic parallelepipedon of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095012.png" />-dimensional lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095013.png" /> that is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095014.png" /> under its logarithmic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095015.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095016.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095017.png" /> is defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095018.png" /> be all real and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095019.png" /> be all pairwise complex non-conjugate isomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095020.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095021.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095022.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095023.png" /> (see [[Dirichlet theorem|Dirichlet theorem]] on units), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095024.png" /> is defined by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095025.png" /></td> </tr></table>
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'' $  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]]; [[Algebraic number theory|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 $
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under its logarithmic mapping  $  l $
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into  $  \mathbf R  ^ {r+} 1 $.
 +
The homomorphism  $  l $
 +
is defined as follows: Let  $  \sigma _ {1} \dots \sigma _ {s} $
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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|Dirichlet theorem]] on units), and  $  l: E \rightarrow \mathbf R  ^ {r+} 1 $
 +
is defined by the formula
 +
 
 +
$$
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l( \alpha )  =  ( l _ {1} ( \alpha ) \dots l _ {s+} t ( \alpha )),
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$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095026.png" /></td> </tr></table>
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$$
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l _ {i} ( \alpha )  = \left \{
  
The image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095027.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095028.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095029.png" />-dimensional lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095030.png" /> lying in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095031.png" /> (where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095032.png" /> are the canonical coordinates).
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The image of $  E $
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under $  l $
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is an r $-
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dimensional lattice in $  \mathbf R  ^ {r+} 1 $
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lying in the plane $  \sum _ {i=} 0  ^ {r+} 1 x _ {i} = 0 $(
 +
where the $  x _ {i} $
 +
are the canonical coordinates).
  
Units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095033.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095034.png" /> form a basis of the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095035.png" /> are known as fundamental units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095036.png" />, and
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Units $  \epsilon _ {1} \dots \epsilon _ {r} $
 +
for which $  l( e _ {1} ) \dots l( e _ {r} ) $
 +
form a basis of the lattice $  l( E) $
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are known as fundamental units of $  K $,  
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and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095037.png" /></td> </tr></table>
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$$
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R _ {K}  = \|  \mathop{\rm det} ( l _ {i} ( \epsilon _ {j} )) _ {i,j= 1 }  ^ {r}
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\| .
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$$
  
There are other formulas linking the regulator with other invariants of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095038.png" /> (see, for example, [[Discriminant|Discriminant]], 3).
+
There are other formulas linking the regulator with other invariants of the field $  K $(
 +
see, for example, [[Discriminant|Discriminant]], 3).
  
If instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095039.png" /> one considers the intersection of this group with an order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095041.png" />, then the regulator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095043.png" /> can be defined in the same way.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1987)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1987)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR></table>

Revision as of 08:10, 6 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

$$ l _ {i} ( \alpha ) = \left \{ The image of $ E $ under $ l $ is an $ r $- dimensional lattice in $ \mathbf R ^ {r+} 1 $ lying in the plane $ \sum _ {i=} 0 ^ {r+} 1 x _ {i} = 0 $( where the $ x _ {i} $ are the canonical coordinates). Units $ \epsilon _ {1} \dots \epsilon _ {r} $ for which $ l( e _ {1} ) \dots l( e _ {r} ) $ form a basis of the lattice $ l( E) $ are known as fundamental units of $ K $, and $$ 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=48493
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article