Octic Hilbert 2-class fields of real quadratic fields with discriminant 8p
Franz Lemmermeyer
TL;DR
The paper addresses constructing explicit cyclic octic Hilbert 2-class fields of real quadratic fields $k=\mathbb{Q}(\sqrt{2p})$ with $p\equiv 1\bmod 8$ and $h_2(k)\equiv 0\bmod 8$. It proceeds by first producing a cyclic quartic unramified extension $K/k$ via $K=k(\sqrt{\alpha})$ with $\alpha=e+f\sqrt{2}$ and $p=e^2-2f^2$, under $e\equiv 3\bmod 4$, $f\equiv 2\bmod 4$, and then lifts to an octic extension by solving the Pell-type equation $eu^2=t^2+2ps^2$ and adjoining $\sqrt{\mu}$ with $\mu=g(u\sqrt{2}+(r+s\sqrt{2})\sqrt{\alpha})$ multiplied by a suitable unit $\varepsilon\in\mathbb{Q}(\sqrt{2})^\times$. Solvability hinges on congruence and norm conditions, and the paper provides explicit instances and a signature-based criterion to realize either totally real or totally complex octic fields via unit choices. The work integrates genus theory, Scholz reciprocity, and explicit diophantine solutions to furnish constructive, verifiable realizations of the desired unramified octic 2-class fields, with clear conditions linking class-number data to the field's real/complex status. The practical impact lies in providing explicit algebraic constructions and verifiable criteria for these high-degree unramified extensions.
Abstract
In this article we explain how to construct cyclic octic unramfied extensions of the real quadratic number field $k = {\mathbb Q}(\sqrt{2p}\,)$, where $p \equiv 1 \bmod 8$ is a prime number such that $h_2(k) \equiv 0 \bmod 8$. The construction only requires solving the diophantine equation $eu^2 = t^2 + 2ps^2$ in integers.