Neuro-Symbolic Learning for Galois Groups: Unveiling Probabilistic Trends in Polynomials

TL;DR

This paper introduces a neurosymbolic method combining neural networks and symbolic reasoning to classify Galois groups of polynomials, revealing new probabilistic trends and distributional insights in algebraic structures.

Contribution

It develops a novel neurosymbolic approach that outperforms traditional methods in classifying Galois groups and uncovers empirical distributional patterns in sextic polynomials.

Findings

Identified 20 sextic polynomials with Galois group C6 across seven invariant classes

Uncovered probabilistic distributional trends in Galois groups under height constraints

Demonstrated AI's ability to reveal patterns beyond traditional symbolic techniques

Abstract

This paper presents a neurosymbolic approach to classifying Galois groups of polynomials, integrating classical Galois theory with machine learning to address challenges in algebraic computation. By combining neural networks with symbolic reasoning we develop a model that outperforms purely numerical methods in accuracy and interpretability. Focusing on sextic polynomials with height $\leq 6$, we analyze a database of 53,972 irreducible examples, uncovering novel distributional trends, such as the 20 sextic polynomials with Galois group $C_6$ spanning just seven invariant-defined equivalence classes. These findings offer the first empirical insights into Galois group probabilities under height constraints and lay the groundwork for exploring solvability by radicals. Demonstrating AI's potential to reveal patterns beyond traditional symbolic techniques, this work paves the way for future research in computational algebra, with implications for probabilistic conjectures and higher degree classifications.