On the main cojecture of Iwasawa theory for imaginary quadratic fields.
The density of primes dividing at least one term of the Lucas sequence , defined by and for , with an arbitrary integer, is determined.
We study the capitulation of -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields , where and are different primes. For each of the three quadratic extensions inside the absolute genus field of , we determine a fundamental system of units and then compute the capitulation kernel of . The generators of the groups and are also determined from which we deduce that is smaller than the relative genus field . Then we prove that each...