Quadratic p-ring spaces for counting dihedral fields

Abstract: 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/C^p) and unit group (U/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 | \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.