From Polynomials to Databases: Arithmetic Structures in Galois Theory

• URL: http://arxiv.org/abs/2511.16622v1
• Date: Thu, 20 Nov 2025 18:29:38 GMT
• Title: From Polynomials to Databases: Arithmetic Structures in Galois Theory
• Authors: Jurgen Mezinaj,
• Abstract summary: We develop a framework for classifying Galois groups of irreducible degree-7s over$mathbbQ$, combining explicit resolvent methods with machine learning techniques.<n>A database of over one million normalizedive project septics is constructed, each annotated with invariants$J_0, dots, J_4$ derived from binary transvections.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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