Difference between revisions of "Regulator of an algebraic number field"
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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 | + | 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 | + | 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+} | + | 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 | + | 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+} | + | l( \alpha ) = ( l _ {1} ( \alpha ) \dots l _ {s+t} ( \alpha )), |
$$ | $$ | ||
| Line 46: | Line 46: | ||
$$ | $$ | ||
l _ {i} ( \alpha ) = \left \{ | 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 . | ||
| + | $$ | ||
The image of $ E $ | The image of $ E $ | ||
under $ l $ | under $ l $ | ||
is an $ r $- | is an $ r $- | ||
| − | dimensional lattice in $ \mathbf R | + | dimensional lattice in $ \mathbf R ^ {r+1} $ |
| − | lying in the plane $ \sum _ {i=} 0 | + | lying in the plane $ \sum _ {i=} 0 ^ {r+1} x _ {i} = 0 $( |
where the $ x _ {i} $ | where the $ x _ {i} $ | ||
are the canonical coordinates). | are the canonical coordinates). | ||
Revision as of 21:46, 7 June 2020
$ K $
The number $ R _ {K} $ that is, by definition, equal to 1 if $ K $ is the field $ \mathbf Q $ or an imaginary quadratic extension of $ \mathbf Q $, and to $ v/ \sqrt {r+1} $ in all other cases, where $ r $ is the rank of the group $ E $ of units of the field $ K $( see Algebraic number; Algebraic number theory) and $ v $ is the $ r $- dimensional volume of the basic parallelepipedon of the $ r $- dimensional lattice in $ \mathbf R ^ {r+1} $ that is the image of $ E $ under its logarithmic mapping $ l $ into $ \mathbf R ^ {r+1} $. The homomorphism $ l $ is defined as follows: Let $ \sigma _ {1} \dots \sigma _ {s} $ be all real and let $ \sigma _ {s+1} \dots \sigma _ {s+t} $ be all pairwise complex non-conjugate isomorphisms of $ K $ into $ \mathbf C $; $ s + 2t = \mathop{\rm dim} _ {\mathbf Q} K $. Then $ {r+1} = {s+t} $( see Dirichlet theorem on units), and $ l: E \rightarrow \mathbf R ^ {r+1} $ is defined by the formula
$$ 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) |
Regulator of an algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regulator_of_an_algebraic_number_field&oldid=48493