  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.
