{"ID":2839210,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.16622","arxiv_id":"2511.16622","title":"From Polynomials to Databases: Arithmetic Structures in Galois Theory","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.","short_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, \\do...","url_abs":"https://arxiv.org/abs/2511.16622","url_pdf":"https://arxiv.org/pdf/2511.16622v1","authors":"[\"Jurgen Mezinaj\"]","published":"2025-11-20T18:29:38Z","proceeding":"math.AC","tasks":"[\"math.AC\",\"cs.LG\"]","methods":"[]","has_code":false}