# From Polynomials to Databases: Arithmetic Structures in Galois Theory

arXiv: https://arxiv.org/abs/2511.16622 · HTML: https://mlsanity.com/paper/2511.16622/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 17 references that are in the map.
- First posted on arXiv 20 Nov 2025.

## Builds on (most cited first, 3 of 3)

- [Machine learning for moduli space of genus two curves and an application to isogeny based cryptography](https://mlsanity.com/paper/2403.17250/) (2024)
- [Graded Neural Networks](https://mlsanity.com/paper/2502.17751/) (2025)
- [Galois groups of polynomials and neurosymbolic networks](https://mlsanity.com/paper/2501.12978/) (2025)

## Abstract (arXiv)

We develop a computational framework for classifying Galois groups of irreducible degree-7 polynomials over~mathbbQ, combining explicit resolvent methods with machine learning techniques. A database of over one million normalized projective septics is constructed, each annotated with algebraic invariants~J0, dots, J4 derived from binary transvections. For each polynomial, we compute resolvent factorizations to determine its Galois group among the seven transitive subgroups of~S7 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.