characterizing CM-fields using Dirichlet’s unit theorem
If is a number field, is the ring of algebraic integers in , and is the (multiplicative) group of units in . Dirichlet’s unit theorem gives the structure of the unit group. We can use that theorem to characterize CM-fields:
We use the notation of the article on Dirichlet’s unit theorem, where (and ) is used to count real embeddings and (as well as ) to count complex embeddings, and we write or for the group of roots of unity in or .
(): If is CM, then since is totally real, . Hence by Dirichlet’s unit theorem, . Since is a complex quadratic extension, and all its embeddings are complex. Thus . Hence as well. Clearly , and since they have the same rank (http://planetmath.org/FreeModule), their quotient is torsion and thus finite.
(): Since is finite, the ranks of these groups are equal and thus again by Dirichlet’s unit theorem.
subtracting (2) from (1), we get
and thus so that . Thus , and since is a nontrivial extension, we must have so that is quadratic and (since ).
Finally, by (3), we then have ; (2) says that , and thus . It follows that is totally real and, since , must be an imaginary quadratic extension of .
|Title||characterizing CM-fields using Dirichlet’s unit theorem|
|Date of creation||2013-03-22 17:57:26|
|Last modified on||2013-03-22 17:57:26|
|Last modified by||rm50 (10146)|