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

arXiv: https://arxiv.org/abs/2502.20844 · HTML: https://mlsanity.com/paper/2502.20844/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 10 references that are in the map.
- First posted on arXiv 28 Feb 2025.

## Builds on (most cited first, 1 of 1)

- [Machine learning for moduli space of genus two curves and an application to isogeny based cryptography](https://mlsanity.com/paper/2403.17250/) (2024)

## Abstract (arXiv)

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 ≤ 6, we analyze a database of 53,972 irreducible examples, uncovering novel distributional trends, such as the 20 sextic polynomials with Galois group C6 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.