By applying interpretable machine learning methods such as decision trees, we study how simple models can classify the Galois groups of Galois extensions over $\mathbb{Q}$ 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.

When you have a training dataframe which contains both number fields and text and apply a simple model from scikit-lean or some equivalent, one of the easiest way is to use sklearn.pipeline.FeatureUnion in Pipeline. The following example assumes X_train to be a pandas DataFrame, which consists of many number fields with a text field at the last column. Then you can create a FunctionTransformer to separate the number columns and the text column. The functions passed to this FunctionTransformer can be anything, so modify this according to your input data. Here it simply return the last column as textual feature and the rest as numerical features.

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.