Hilbert 2-Class Fields and 2-Descent

TL;DR

This paper constructs specific unramified cyclic octic extensions of complex quadratic fields using binary quadratic forms, linking them to 2-descents on Pell conics and elliptic curves, and describing cyclic quartic extensions explicitly.

Contribution

It introduces a new construction method for unramified cyclic octic extensions and connects it to 2-descent theory and explicit descriptions of cyclic quartic extensions.

Findings

Construction of unramified cyclic octic extensions for certain quadratic fields

Connection between quadratic forms and 2-descents on elliptic curves

Explicit description of cyclic quartic extensions

Abstract

We give a construction of unramified cyclic octic extensions of certain complex quadratic number fields. The binary quadratic form used in this construction also shows up in the theory of 2-descents on Pell conics and elliptic curves, as well as in the explicit description of cyclic quartic extensions.