# Machines Learn Number Fields, But How? The Case of Galois Groups

arXiv: https://arxiv.org/abs/2508.06670 · HTML: https://mlsanity.com/paper/2508.06670/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 44 references that are in the map.
- First posted on arXiv 8 Aug 2025.

## Builds on (most cited first, 6 of 6)

- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [Machine-Learning Arithmetic Curves](https://mlsanity.com/paper/2012.04084/) (2020)
- [Machine-Learning Number Fields](https://mlsanity.com/paper/2011.08958/) (2020)
- [Machine Learning Class Numbers of Real Quadratic Fields](https://mlsanity.com/paper/2209.09283/) (2022)
- [Mathematical Data Science](https://mlsanity.com/paper/2502.08620/) (2025)

## Built on by (most cited first, 2 of 2)

- [Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves](https://mlsanity.com/paper/2603.17681/) (2026)
- [Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves](https://mlsanity.com/paper/2607.24251/) (2026)

## Abstract (arXiv)

By applying interpretable machine learning methods such as decision trees, we study how simple models can classify the Galois groups of Galois extensions over mathbbQ of degrees 4, 6, 8, 9, and 10, using Dedekind zeta coefficients. Our interpretation of the machine learning results allows us to understand how the distribution of zeta coefficients depends on the Galois group, and to prove new criteria for classifying the Galois groups of these extensions. Combined with previous results, this work provides another example of a new paradigm in mathematical research driven by machine learning.