Galois groups of polynomials and neurosymbolic networks

TL;DR

This paper proposes a neurosymbolic network approach to classify Galois groups of polynomials, improving efficiency over traditional neural networks and revealing new polynomial group distributions.

Contribution

It introduces a novel neurosymbolic network method for classifying Galois groups, enhancing computational efficiency and uncovering new polynomial group distributions.

Findings

Neurosymbolic networks outperform standard neural networks in classifying Galois groups.

Discovered unique polynomial distributions for non-symmetric, non-alternating groups.

Method demonstrates potential for broader applications in algebra and machine learning.

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.