Principalization algorithm via class group structure

Abstract

For an algebraic number field K with 3-class group \(Cl_3(K)\) of type (3,3), the structure of the 3-class groups \(Cl_3(N_i)\) of the four unramified cyclic cubic extension fields \(N_i\), \(1\le i\le 4\), of K is calculated with the aid of presentations for the metabelian Galois group \(G_3^2(K)=Gal(F_3^2(K) | K)\) of the second Hilbert 3-class field \(F_3^2(K)\) of K. In the case of a quadratic base field \(K=\mathbb{Q}(\sqrt{D})\) it is shown that the structure of the 3-class groups of the four \(S_3\)-fields \(N_1,\ldots,N_4\) frequently determines the type of principalization of the 3-class group of K in \(N_1,\ldots,N_4\). This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all 4596 quadratic fields K with 3-class group of type (3,3) and discriminant \(-10^6<D<10^7\) to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups \(G_3^2(K)\) on various coclass trees of the coclass graphs G(3,r), \(1\le r\le 6\), in the sense of Eick, Leedham-Green, and Newman.