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 r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet M(K(f))=[(N(c,i))_{1<=i<=m(c)}]_{c|f} of dihedral fields N(c,i) with various conductors c|f having p-multiplicities m(c) over K such that sum_{c|f} m(c)=(p^r(f)-1)/(p-1). The advanced viewpoint of classifying the entire collection M(K(f)), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields, and the actual construction of the multiplet M(K(f)) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.