🤖 AI Summary
This work addresses the classification of Galois groups of irreducible septic polynomials over the rationals. We propose a neuro-symbolic hybrid method: leveraging algebraic invariants $J_0$–$J_4$ derived from binary transvectants, combined with explicit factorization, symbolic computation, and supervised learning, to construct the first million-scale database of normalized projected septic polynomials, each annotated with its exact Galois group. Our framework achieves high-accuracy identification of all seven transitive subgroups of $S_7$, notably improving discrimination of rare solvable groups. The database enables empirical analysis of subgroup distributions under constrained conditions. The resulting neuro-symbolic classifier balances interpretability and generalizability, and the methodology is extensible to higher-degree polynomials—advancing computational and data-driven research in constructive 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.