Mathematics, Computation and Geometry of Data

Volume 2 (2022)

Number 1

Machine-learning number fields

Pages: 49 – 66

DOI: https://dx.doi.org/10.4310/MCGD.2022.v2.n1.a2

Authors

Yang-Hui He (Department of Mathematics, City University of London, United Kingdom; Merton College, University of Oxford, United Kingdom; and School of Physics, NanKai University, Tianjin, China)

Kyu-Hwan Lee (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Thomas Oliver (SCEDT, Teesside University, Middlesbrough, United Kingdom)

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 $\gt 0.97$.

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Y.H.H. is indebted to STFC UK, for grant ST/J00037X/1.

K.H.L. is partially supported by a grant from the Simons Foundation (#712100).

T.O. acknowledges support from the EPSRC through research grant EP/S032460/1.

Received 7 March 2021

Published 21 October 2022