English

From Polynomials to Databases: Arithmetic Structures in Galois Theory

Commutative Algebra 2025-11-21 v1 Machine Learning

Abstract

We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~Q\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~J0,,J4J_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~S7S_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.

Keywords

Cite

@article{arxiv.2511.16622,
  title  = {From Polynomials to Databases: Arithmetic Structures in Galois Theory},
  author = {Jurgen Mezinaj},
  journal= {arXiv preprint arXiv:2511.16622},
  year   = {2025}
}
R2 v1 2026-07-01T07:47:47.674Z