From Polynomials to Databases: Arithmetic Structures in Galois Theory

TL;DR

This paper introduces a computational framework combining algebraic invariants, resolvent methods, and machine learning to classify Galois groups of degree-7 polynomials, creating a large database and improving detection accuracy.

Contribution

It develops a hybrid symbolic-numeric approach for classifying Galois groups, constructs a comprehensive database, and demonstrates enhanced machine learning techniques for algebraic invariants.

Findings

Created a database of over one million degree-7 polynomials with Galois group annotations

Developed a neurosymbolic classifier that outperforms coefficient-based models

Extended the methodology to higher-degree polynomials and complex algebraic structures

Abstract

We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~$\mathbb{Q}$, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~$J_0, \dots, J_4$ derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~$S_7$ identified by Foulkes. Using this dataset, we train a neurosymbolic classifier that integrates invariant-theoretic features with supervised learning, yielding improved accuracy in detecting rare solvable groups compared to coefficient-based models. The resulting database provides a reproducible resource for constructive Galois theory and supports empirical investigations into group distribution under height constraints. The methodology extends to higher-degree cases and illustrates the utility of hybrid symbolic-numeric techniques in computational algebra.