On Hilbert-Speiser type imaginary quadratic fields
Let be a totally real algebraic number field whose ring of integers is a principal ideal domain. Let be a totally definite ternary quadratic form with coefficients in . We shall study representations of totally positive elements by . We prove a quantitative formula relating the number of representations of by different classes in the genus of to the class number of , where is a constant depending only on . We give an algebraic proof of a classical result of H. Maass on representations...
Let be a square free integer and . In the present work we determine all the fields such that the -class group, , of is of type or .
Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition , where is the second Hilbert 2-class field of k.
Let be an odd square-free integer, any integer and . In this paper, we shall determine all the fields having an odd class number. Furthermore, using the cyclotomic -extensions of some number fields, we compute the rank of the -class group of whenever the prime divisors of are congruent to or .