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 [Formula: see text], p-ring spaces Vp(c) modulo c are introduced by defining a morphism ψ : f ↦ Vp(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product Vp of the p-elementary class group 𝒞/𝒞p and unit group U/Up 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 | ℚ) and sharing a common discriminant dN and conductor c over K. The number mp(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 Vp over the finite field 𝔽p, called the modified p-class rank σp of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σp ≤ 1 only. Here, the results are extended to σp = 2, underpinned by concrete numerical examples.