Dihedral side extensions and class groups
Let \(p\) be an odd prime, let \(L/\mathbb Q\) be an extension of degree \(p\) whose normal closure \(K/\mathbb Q\) has Galois group \(D_p\), the dihedral group of order \(2p\), and let \(F\) denote the unique quadratic subfield of \(K\). The author proves that if there is exactly one ramified prime in \(K/F\), then the ranks of the \(p\)-class groups of \(F\) and \(L\) are related by the inequalities \[ \text{rk\;Cl}_p(F) \leq \text{rk\;Cl}_p(L) \leq \frac{p-1}2 \text{rk\;Cl}_p(F). \] This generalizes a result due to \textit{T. Callahan} [Mathematika 21(1974), 168--188 (1975; Zbl 0312.12009)] according to which the ranks of the \(3\)-class groups of \(F\) and \(L\) coincide. Results similar to those proved here were derived by \textit{R. Bölling} [Math. Nachr. 135, 275--310 (1988; Zbl 0674.12003)] in the case of unramified extensions, which in turn were generalized by the reviewer [Math. Nachr. 278, No. 6, 679--691 (2005; Zbl 1067.11069)] to general base fields with class number not divisible by \(p\).
scientific article; zbMATH DE number 5378822