Machines Learn Number Fields, But How? The Case of Galois Groups

TL;DR

This paper demonstrates how interpretable machine learning, specifically decision trees, can classify Galois groups of number field extensions over  with high accuracy, revealing new criteria and insights into their distribution.

Contribution

It introduces a novel approach using decision trees to classify Galois groups based on Dedekind zeta coefficients, providing interpretability and new classification criteria.

Findings

Decision trees can accurately classify Galois groups of degree 4, 6, 8, 9, and 10 extensions.

The study uncovers how zeta coefficient distributions depend on Galois groups.

New criteria for Galois group classification are proven using machine learning insights.

Abstract

By applying interpretable machine learning methods such as decision trees, we study how simple models can classify the Galois groups of Galois extensions over $\mathbb{Q}$ of degrees 4, 6, 8, 9, and 10, using Dedekind zeta coefficients. Our interpretation of the machine learning results allows us to understand how the distribution of zeta coefficients depends on the Galois group, and to prove new criteria for classifying the Galois groups of these extensions. Combined with previous results, this work provides another example of a new paradigm in mathematical research driven by machine learning.