Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors

Abstract

Each \(p\)-ring class field \(K_f\) modulo a \(p\)-admissible conductor \(f\) over a quadratic base field \(K\) with \(p\)-ring class rank \(\varrho_f\) mod \(f\) is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet \(\mathbf{M}(K_f)=\lbrack(N_{c,i})_{1\le i\le m(c)}\rbrack_{c\mid f}\) of dihedral fields \(N_{c,i}\) with various conductors \(c\mid f\) having \(p\)-multiplicities \(m(c)\) over \(K\) such that \(\sum_{c\mid f}\,m(c)=\frac{p^{\varrho_f}-1}{p-1}\). The advanced viewpoint of classifying the entire collection \(\mathbf{M}(K_f)\), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields. The actual construction of the multiplet \(\mathbf{M}(K_f)\) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.