Let \(K\) be a quadratic number field of discriminant \(d\). For an odd prime \(p\) let \(m_p(d,c)\) denote the \(p\)-multiplicity of \(c\) with respect to \(d\), the number of degree \(p\) extensions of \(K\) with conductor \(c\) and absolute Galois group isomorphic to the dihedral group of order \(2p\). The author examines this quantity in detail, using an ideal-theoretic setup to unify several existing results and to solve new cases. Many interesting tables of data and concrete examples are given that illustrate the main ideas in detail. The author provides examples of interest in statistical algebraic number theory such as fields of small discriminant with given \(p\)-multiplicity. These tables correct some earlier errors that have appeared in the literature.
The author makes a convincing case that much of the difficult of computing \(m_p(d,c)\) is governed by the ‘\(p\)-defect’ \(\delta_p(c)\) with respect to \(K\). This \(p\)-defect is the codimension of the ‘\(p\)-ring space’ \(V_p(c)\) in the \(\mathbb{F}_p\)-vector space \(V_p\) of nontrivial \(p\)th powers of ideals. Previously formulas for the \(p\)-mulitplicity were known only for codimension \(0\) and \(1\), and here these are extended to codimension \(2\). In particular, the author builds on his earlier work [Math. Comput. 58, No. 198, 831–848 (1992; Zbl 0737.11028)]. These definitions are laid out clearly and a helpful picture illustrating the \(p\)-defect is given in Figure 1. Special care must be taken to deal with \(m_3(d,c)\) when \(c\) is an irregular conductor, where \(d \equiv -3 \mod 9\) and \(9\) divides \(c\), and the author nicely unifies this case with the more general setting.