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

2025-08-08

AI Summary

This study addresses the problem of establishing interpretable connections between coefficients of Dedekind zeta functions and the Galois group structures of low-degree (4, 6, 8, 9, 10) Galois extensions over ℚ. Using interpretable machine learning—particularly decision trees—we perform feature engineering on zeta coefficient sequences and build classification models to systematically identify discriminative coefficient patterns characteristic of distinct Galois groups. Our key contributions are twofold: first, achieving high-accuracy group classification (>95% accuracy); second, extracting concise, mathematically verifiable classification criteria—e.g., sign patterns and modular constraints on initial zeta coefficients—that yield data-driven insights into classical number-theoretic questions. This work advances an interdisciplinary paradigm bridging interpretable AI and algebraic number theory, providing a generalizable methodological framework for L-function inverse problems and explicit classification theories.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Classify Galois groups using machine learning methods
• Understand zeta coefficients' dependence on Galois groups
• Prove new criteria for Galois group classification

Innovation

• Interpretable machine learning for Galois groups
• Decision trees classify zeta coefficients
• New criteria from machine learning results