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From Polynomials to Databases: Arithmetic Structures in Galois Theory

Published: November 20, 2025 | arXiv ID: 2511.16622v1

By: Jurgen Mezinaj

Potential Business Impact:

Helps math computers find hidden number patterns faster.

Business Areas:
Database Data and Analytics, Software

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.

Page Count
33 pages

Category
Mathematics:
Commutative Algebra