Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Polynomials to Databases: Arithmetic Structures in Galois Theory

Jurgen Mezinaj

TL;DR

This work presents a scalable framework for classifying Galois groups of irreducible degree-7 polynomials over $\mathbb{Q}$ by integrating classical resolvent techniques with neurosymbolic machine learning. It builds a large, reproducible database of over $1.18$ million septics annotated with invariant features and discriminants, and uses targeted resolvents (quadratic, 30-ic, 120-ic, and 35-ic) to identify the seven transitive subgroups of $S_7$ as classified by Foulkes. A neural-network pipeline augmented with invariant-based features improves detection of rare solvable groups, with non-$S_7$ analyses yielding notable gains in balanced accuracy. The work also connects constructive Galois theory via Hurwitz spaces and Hilbert irreducibility, providing explicit polynomials realizing groups such as $C_7$ and $C_7\rtimes C_3$, and discusses extensions to higher degrees. Overall, the paper demonstrates how hybrid symbolic-numeric methods can address the inverse Galois problem and enable empirical studies of Galois-group distributions under height constraints.

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.

From Polynomials to Databases: Arithmetic Structures in Galois Theory

TL;DR

This work presents a scalable framework for classifying Galois groups of irreducible degree-7 polynomials over by integrating classical resolvent techniques with neurosymbolic machine learning. It builds a large, reproducible database of over million septics annotated with invariant features and discriminants, and uses targeted resolvents (quadratic, 30-ic, 120-ic, and 35-ic) to identify the seven transitive subgroups of as classified by Foulkes. A neural-network pipeline augmented with invariant-based features improves detection of rare solvable groups, with non- analyses yielding notable gains in balanced accuracy. The work also connects constructive Galois theory via Hurwitz spaces and Hilbert irreducibility, providing explicit polynomials realizing groups such as and , and discusses extensions to higher degrees. Overall, the paper demonstrates how hybrid symbolic-numeric methods can address the inverse Galois problem and enable empirical studies of Galois-group distributions under height constraints.

Abstract

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

Paper Structure

This paper contains 49 sections, 9 theorems, 68 equations, 9 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Galois theory and septics polynomials
  3. Galois groups of irreducible polynomials
  4. Septic polynomials
  5. Invariants of septics
  6. Invariants of septics
  7. Resolvents
  8. Express Roots Symbolically
  9. Compute Power Sums
  10. Apply Newton’s Identities
  11. Construct the Resolvent
  12. Resolvents of septics
  13. The Quadratic Resolvent
  14. The 30-ic Resolvent for $\mathop{\mathrm{PSL}}\nolimits(3,2)$
  15. The 120-ic Resolvent for $C_7 \rtimes C_6$
  16. ...and 34 more sections

Key Result

Theorem 2.1

Let $E/F$ be a finite Galois extension with Galois group $G = \mathop{\mathrm{Gal}}\nolimits(E/F)$. Then:

Figures (9)

  • Figure 1: The lattice of subgroups of $S_7$
  • Figure 2: Relative frequencies of the six non-$S_7$ Galois groups.
  • Figure 3: Empirical distributions of the log-height $h(f)$ for each non-$S_7$ Galois group.
  • Figure 4: Boxplots of $h(f)$ by Galois group. Each group occupies its own characteristic height range.
  • Figure 5: Mean log-height $\mathbb{E}[h(f)\mid G]$ for each non-$S_7$ Galois group.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Theorem 2.1: Fundamental Theorem of Galois Theory
  • proof
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 7 more