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$#C+1 = 43 : ~/encyclopedia/old_files/data/R080/R.0800950 Regulator of an algebraic number field
 
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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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'' $  K $''
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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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The number  $  R _ {K} $
 
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that is, by definition, equal to 1 if  $  K $
 
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is the field  $  \mathbf Q $
 
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or an imaginary quadratic extension of  $  \mathbf Q $,
 
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and to  $  v/ \sqrt r+ 1 $
 
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in all other cases, where  $  r $
 
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is the rank of the group  $  E $
 
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of units of the field  $  K $(
 
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see [[Algebraic number|Algebraic number]]; [[Algebraic number theory|Algebraic number theory]]) and  $  v $
 
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is the  $  r $-
 
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dimensional volume of the basic parallelepipedon of the  $  r $-
 
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dimensional lattice in  $  \mathbf R  ^ {r+} 1 $
 
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that is the image of  $  E $
 
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under its logarithmic mapping  $  l $
 
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into  $  \mathbf R  ^ {r+} 1 $.
 
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The homomorphism  $  l $
 
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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 $
 
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be all pairwise complex non-conjugate isomorphisms of  $  K $
 
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into  $  \mathbf C $;  
 
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$  s + 2t = \mathop{\rm dim} _ {\mathbf Q}  K $.
 
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Then  $  r+ 1 = s+ t $(
 
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see [[Dirichlet theorem|Dirichlet theorem]] on units), and  $  l: E \rightarrow \mathbf R  ^ {r+} 1 $
 
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is defined by the formula
 
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$$
 
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l( \alpha )  =  ( l _ {1} ( \alpha ) \dots l _ {s+} t ( \alpha )),
 
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$$
 
  
 
where
 
where
  
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$$
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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/r08095026.png" /></td> </tr></table>
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l _ {i} ( \alpha )  = \left \{
 
  
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The image of $  E $
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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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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 $(
 
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where the $  x _ {i} $
 
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are the canonical coordinates).
 
  
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Units $  \epsilon _ {1} \dots \epsilon _ {r} $
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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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for which $  l( e _ {1} ) \dots l( e _ {r} ) $
 
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form a basis of the lattice $  l( E) $
 
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are known as fundamental units of $  K $,  
 
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and
 
  
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$$
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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/r08095037.png" /></td> </tr></table>
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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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$$
 
  
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There are other formulas linking the regulator with other invariants of the field $  K $(
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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).
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see, for example, [[Discriminant|Discriminant]], 3).
 
  
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If instead of $  E $
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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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one considers the intersection of this group with an order $  {\mathcal O} $
 
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of $  K $,  
 
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then the regulator $  R _  {\mathcal O}  $
 
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of $  {\mathcal O} $
 
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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 14:53, 7 June 2020

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)
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
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