Quadratic fields with cyclic 2-class groups

TL;DR

This paper proves that for any positive integer k, there are infinitely many complex quadratic fields with cyclic 2-class groups of order 2^k, using a combination of analytic and algebraic methods.

Contribution

It introduces a novel approach combining the circle method with an algebraic criterion to establish the existence of infinitely many such fields.

Findings

Infinitely many complex quadratic fields have cyclic 2-class groups of order 2^k.

The proof applies the circle method to number field problems.

An algebraic criterion for ideal classes to be squares is developed.

Abstract

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square.