regulator
Let be a number field with . Here denotes the number of real embeddings:
while is half of the number of complex embeddings:
Note that are all the complex embeddings of . Let and for define the “norm” in corresponding to each embedding:
Let be the ring of integers of . By Dirichlet’s unit theorem, we know that the rank of the unit group is exactly . Let
be a fundamental system of generators of modulo roots of unity (this is, modulo the torsion subgroup). Let be the matrix
and let be the matrix obtained by deleting the -th row from , . It can be checked that the determinant of , , is independent up to sign of the choice of fundamental system of generators of and is also independent of the choice of .
Definition.
The regulator of is defined to be
The regulator is one of the main ingredients in the analytic class number formula for number fields.
References
- 1 Daniel A. Marcus, Number Fields, Springer, New York.
- 2 Serge Lang, Algebraic Number Theory. Springer-Verlag, New York.
Title | regulator |
---|---|
Canonical name | Regulator |
Date of creation | 2013-03-22 13:54:34 |
Last modified on | 2013-03-22 13:54:34 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 8 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 11R27 |
Related topic | NumberField |
Related topic | DirichletsUnitTheorem |
Related topic | ClassNumberFormula |
Related topic | RegulatorOfAnEllipticCurve |
Defines | regulator of a number field |