# The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fields

## TL;DR

This paper develops a pattern recognition strategy using Artin transfers to determine the Galois group structure of 2-class field towers in quadratic fields with specific class group invariants, advancing understanding of their tower lengths.

## Contribution

It introduces a novel approach combining Artin pattern analysis with p-group generation to classify metabelian 2-groups in 2-class field towers, extending existing classifications.

## Key findings

- Determined Galois group types for specific quadratic fields.
- Linked tower length to 2-class group rank.
- Enhanced classification methods using pattern recognition.

## Abstract

The isomorphism type of the Galois group of the 2-class field tower of quadratic number fields having a 2-class group with abelian type invariants (4,4) is determined by means of information on the transfer of 2-classes to unramified abelian 2-extensions, collected in the Artin pattern. In recent investigations by Benjamin and Snyder, the length of the tower of such fields has turned out to be dependent on the rank of the 2-class group of the first Hilbert 2-class field. Significant progress is achieved by extending the pool of possible metabelian 2-groups of the second Hilbert 2-class field from the SmallGroups database, resp. Hall-Senior classification, with the aid of the p-group generation algorithm, and sifting the pool by means of pattern recognition.