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Regulator of an algebraic number field

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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. V.L. Popov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regulator_of_an_algebraic_number_field&oldid=17455
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098