Difference between revisions of "Regulator of an algebraic number field"
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− | < | + | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r0809501.png" />'' |
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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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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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− | The image of | + | 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). |
− | under | ||
− | is an | ||
− | dimensional lattice in | ||
− | lying in the plane | ||
− | where the | ||
− | are the canonical coordinates). | ||
− | Units | + | 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 |
− | for which | ||
− | form a basis of the lattice | ||
− | are known as fundamental units of | ||
− | 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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− | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080950/r08095038.png" /> (see, for example, [[Discriminant|Discriminant]], 3). |
− | see, for example, [[Discriminant|Discriminant]], 3). | ||
− | If instead of | + | 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. |
− | one considers the intersection of this group with an order | ||
− | of | ||
− | then the regulator | ||
− | of | ||
− | 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) |
Regulator of an algebraic number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regulator_of_an_algebraic_number_field&oldid=48493