The second p-class group of a number field

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.

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