Quadratic p-ring spaces for counting dihedral fields

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field $K=\mathbb{Q}(\sqrt{d})$, p-ring spaces $V_p(c)$ modulo c are introduced by defining a morphism $\psi :f\mapsto V_p(f)$ from the divisor lattice $\mathbb{N}$ of positive integers to the lattice S of subspaces of the direct product $V_p$ of the p-elementary class group $C\mathrm{/}{C}^p$ and unit group $U\mathrm{/}{U}^p$ of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group $Gal(N\mathrm{\mid }\mathbb{Q})$ and sharing a common discriminant $d_N$ and conductor c over K. The number $m_p(d,c)$ of these extensions is given by a formula in terms of positions of p-ring spaces in S, whose complexity increases with the dimension of the vector space $V_p$ over the finite field ${\mathbb{F}}_p$, called the modified p-class rank ${\sigma }_p$ of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with $0\le {\sigma }_p\le 1$ only. Here, the results are extended to ${\sigma }_p=2$, underpinned by concrete numerical examples.