Galois groups of polynomials and neurosymbolic networks

TL;DR

This work addresses predicting Galois groups and solvability by radicals for polynomials using a neurosymbolic framework that combines neural and symbolic reasoning. It builds large, invariant-rich databases of irreducible polynomials up to quintic, annotated with discriminants, moduli heights, and other invariants, and leverages reduction modulo primes and real-root counts to constrain candidates. A GaloisNetwork architecture integrates symbolic layers (real-root counts, modular signatures, discriminant checks) with neural classification to achieve interpretable, high-accuracy predictions. The study demonstrates both practical datasets and theoretical insights, pointing to scalable improvements for higher-degree polynomials and broader connections with algebraic geometry and AI-driven mathematics.

Abstract

This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline the process of determining solvability by radicals and explore broader applications within Galois theory. This summary encapsulates the background, methodology, potential applications, and challenges of using data science in Galois theory. More specifically, we design a neurosymbolic network to classify Galois groups and show how this is more efficient than usual neural networks. We discover some very interesting distribution of polynomials for groups not isomorphic to the symmetric groups and alternating groups.

Figures (7)

• Figure 1: Lattice of transitive subgroups of $S_5$
• Figure 2: Distribution of cubics with Galois group $C_3$.
• Figure 3: Occurrences for cubics versus the invariants
• Figure 4: Distribution of quartics with $\hbox{Gal} (f) \not\cong S_4$.
• Figure 5: The ratio of weighted height with naive height

Theorems & Definitions (16)

• Lemma 1
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• Lemma 10