Imaginary quadratic fields with 2-class group of type $(2,2^\ell)$

TL;DR

This paper proves the infinitude of imaginary quadratic fields with specific 2-class group structures and provides bounds on their count, extending previous cyclic 2-class group results using advanced number theory techniques.

Contribution

It establishes the existence of infinitely many such fields for any positive integer ll and introduces new congruence conditions into existing number theory results.

Findings

Infinitely many imaginary quadratic fields with 2-class group of type (2, 2^ll) exist.

Provides a lower bound for the number of these fields with bounded discriminant.

Extends previous results on cyclic 2-class groups using congruence conditions.

Abstract

We prove that for any given positive integer $\ell$ there are infinitely many imaginary quadratic fields with 2-class group of type $(2,2^\ell)$, and provide a lower bound for the number of such groups with bounded discriminant for $\ell\ge2$. This work is based on a related result for cyclic 2-class groups by Dominguez, Miller and Wong, and our proof proceeds similarly. Our proof requires introducing congruence conditions into Perelli's result on Goldbach numbers represented by polynomials, which we establitish in some generality.