The second p-class group of a number field

Abstract: For a prime (p\ge 2) and a number field K with p-class group of type (p,p) it is shown that the class, coclass, and further invariants of the metabelian Galois group (G=Gal(F_p^2(K) | K)) of the second Hilbert p-class field (F_p^2(K)) of K are determined by the p-class numbers of the unramified cyclic extensions (N_i | K), (1\le i\le p+1), of relative degree p. In the case of a quadratic field (K=\mathbb{Q}(\sqrt{D})) and an odd prime (p\ge 3), the invariants of G are derived from the p-class numbers of the non-Galois subfields (L_i | \mathbb{Q}) of absolute degree p of the dihedral fields (N_i). As an application, the structure of the automorphism group (G=Gal(F_3^2(K) | K)) of the second Hilbert 3-class field (F_3^2(K)) is analysed for all quadratic fields K with discriminant (-10^6<D<10^7) and 3-class group of type (3,3) by computing their principalisation types. The distribution of these metabelian 3-groups G on the coclass graphs G(3,r), (1\le r\le 6), in the sense of Eick and Leedham-Green is investigated.