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.