2014/03/15 by Daniel C. Mayer, Mayer, Daniel C. · 2 citations
Mathematics · #11R11 #11R16 #11R20 #11R29 #20D15 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R11 #msc:11R16 #msc:11R20 #msc:11R29 #msc:20D15
paper · pdf · doi:10.48550/arxiv.1403.3839
33 pages, 2 figures, presented at the Joint CSASC Conference, Danube University, Krems, Austria, September 2011
arxiv created 2014/03/15 · arxiv updated 2014/03/18
For an algebraic number field K with 3-class group \(Cl3(K)\) of type (3,3), the structure of the 3-class groups \(Cl3(Ni)\) of the four unramified cyclic cubic extension fields \(Ni\), \(1≤ i≤ 4\), of K is calculated with the aid of presentations for the metabelian Galois group \(G32(K)=Gal(F32(K) | K)\) of the second Hilbert 3-class field \(F32(K)\) of K. In the case of a quadratic base field \(K=ℚ(√(D))\) it is shown that the structure of the 3-class groups of the four \(S3\)-fields \(N1,…,N4\) frequently determines the type of principalization of the 3-class group of K in \(N1,…,N4\). 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 \(-106<D<107\) to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups \(G32(K)\) on various coclass trees of the coclass graphs G(3,r), \(1≤ r≤ 6\), in the sense of Eick, Leedham-Green, and Newman.