From Polynomials to Databases: Arithmetic Structures in Galois Theory
This provides a reproducible resource for computational algebra researchers, though it appears incremental as it extends known methods with machine learning integration.
The authors developed a computational framework to classify Galois groups of irreducible degree-7 polynomials over rational numbers, creating a database of over one million polynomials and training a neurosymbolic classifier that improved accuracy in detecting rare solvable groups compared to existing models.
We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~$\mathbb{Q}$, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~$J_0, \dots, J_4$ derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~$S_7$ 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.