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.
翻译:我们开发了一个计算框架,用于分类有理数域上不可约七次多项式的伽罗瓦群,将显式预解式方法与机器学习技术相结合。构建了一个包含超过一百万归一化投影七次多项式的数据库,每个多项式均标注有从二元转置导出的代数不变量~$J_0, \dots, J_4$。针对每个多项式,我们计算预解式因式分解以确定其伽罗瓦群,该群属于Foulkes所识别的~$S_7$的七个可迁子群之一。利用该数据集,我们训练了一个神经符号分类器,该分类器将不变量理论特征与监督学习相融合,相比基于系数的模型,在检测罕见可解群方面实现了更高的准确率。所得数据库为构造性伽罗瓦理论提供了可复现的资源,并支持在高度约束下对群分布进行实证研究。该方法可推广至高次情形,并展示了混合符号-数值技术在计算代数中的实用性。