Difference between revisions of "Regulator of an algebraic number field"
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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 $ | ||
| + | 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|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 | 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 | + | 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 | + | 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 | + | There are other formulas linking the regulator with other invariants of the field $ K $( |
| + | see, for example, [[Discriminant|Discriminant]], 3). | ||
| − | If instead of | + | 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> | ||
Latest revision as of 10:48, 20 January 2024
$ 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=49399
Regulator of an algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regulator_of_an_algebraic_number_field&oldid=49399
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article