Imaginary quadratic fields k with Cl_2(k) = (2,2^m) and Rank Cl_2(k^1) = 2

TL;DR

This paper classifies complex quadratic fields with specific 2-class group structures and confirms their 2-class field towers have length 2, providing insights into the structure of such number fields.

Contribution

The paper provides a complete classification of complex quadratic fields with 2-class group of type (2,2^m) and 2-rank of their Hilbert 2-class fields equal to 2, and establishes their class tower length.

Findings

All such fields have class tower length 2.

The smallest candidate for a length 3 tower is the field with discriminant -1015.

The classification covers all complex quadratic fields with the specified 2-class group structure.

Abstract

We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of fields with 2-class field tower of length 3, but the smallest candidate is the field with discriminant -1015.