Generalized Artin pattern of heterogeneous multiplets of dihedral fields and proof of Scholz's conjecture

Abstract

The concept of Artin transfer pattern $((\ker (T_{K,N_i}){)}_i,({\mathrm{Cl}}_p(N_i){)}_i)$ for homogeneous multiplets $(N_1,\dots ,N_m)$ of unramified cyclic prime degree p extensions $N_i/K$ of a base field K with p-class transfer homomorphisms$T_{K,N_i}:{\mathrm{Cl}}_p(K)\to {\mathrm{Cl}}_p(N_i)$ is generalized for heterogeneous multiplets of ramified extensions. By application to quadratic subfields K of dihedral fields N of degree 2p with an odd prime p, a conjecture of Scholz concerning the index of subfield units, $(U_N:U_0)$, for ramified extensions N/K with conductor f>1 is verified computationally.

18 pages, 12 figures, 5 tables, International Conference on Algebra, Number Theory and Applications (ICANTA) 2019, Oujda, Morocco

References in corpus (1)

• Transfers of metabelian p-groups