Machine-Learning Number Fields

TL;DR

This paper demonstrates that machine-learning algorithms can accurately predict invariants of algebraic number fields, such as class numbers and Galois group structures, even extrapolating beyond the training data.

Contribution

It introduces machine-learning models trained on number field invariants to predict properties like class number and Galois groups with high accuracy, showcasing novel applications in algebraic number theory.

Findings

Random-forest classifier distinguishes real quadratic fields by class number with 96% precision.

Models extrapolate to discriminants outside training range.

Logistic regression predicts Galois groups and unit group ranks with over 97% precision.

Abstract

We show that standard machine-learning algorithms may be trained to predict certain invariants of algebraic number fields to high accuracy. A random-forest classifier that is trained on finitely many Dedekind zeta coefficients is able to distinguish between real quadratic fields with class number 1 and 2, to 0.96 precision. Furthermore, the classifier is able to extrapolate to fields with discriminant outside the range of the training data. When trained on the coefficients of defining polynomials for Galois extensions of degrees 2, 6, and 8, a logistic regression classifier can distinguish between Galois groups and predict the ranks of unit groups with precision >0.97.