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:
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 .
The regulator of is defined to be
The regulator is one of the main ingredients in the analytic class number formula for number fields.
- 1 Daniel A. Marcus, Number Fields, Springer, New York.
- 2 Serge Lang, Algebraic Number Theory. Springer-Verlag, New York.
|Date of creation||2013-03-22 13:54:34|
|Last modified on||2013-03-22 13:54:34|
|Last modified by||alozano (2414)|
|Defines||regulator of a number field|