Coclass of Gal(k(2)((2))/k) for some fields k = Q(root p1p2q,root-1) with 2-class groups of types (2,2,2)

摘要

Let p1 equivalent to p2 equivalent to - q equivalent to 1 (mod 4) be primes such that (2/p1) = 1 and (2/p2) = (p1/p2) = (p1/q) = -1. Put i = root-1 and d = p1p2q, then the bicyclic biquadratic field k = Q(root d, i) has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group Gal(k(2)((2))/k) of the second Hilbert 2-class field k(2)((2)) of k, we study the 2-class field tower of k, and we study the capitulation problem of the 2-classes of k in its fourteen abelian unramified extensions of relative degrees two and four.