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

Each p -ring class field K f modulo a p -admissible conductor f over a quadratic base field K with p -ring class rank ϱ 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 ∑ c ∣ f m ( c ) = p ϱ 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. The actual construction of the multiplet M ( K f ) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.