The k-nearest neighbors (kNN) algorithm is fundamentally grounded on a notion of distance thatdefines similarity and governs decision boundaries. While most applications rely on Euclidean or Riemannianmetrics, these approaches implicitly assume a continuous geometric structure of the data space. In this work, weexplore an alternative and purely algebraic perspective: the use of Galois symmetries to induce a distancesuitable for kNN-based decision making. This work is proposing a distance defined via embeddings generated bythe Galois group of a field extension, ensuring invariance under algebraic conjugation. A fully explicit numericalexample over a quadratic field extension illustrates the method and its integration into a kNN classifier.Although the proposed distance is not intended as a general-purpose replacement for geometric metrics, ithighlights a novel connection between algebraic symmetry, metric construction, and decision theory, opening aconceptual pathway toward symmetry-aware learning algorithms.
Rodolfo Moroz (Thu,) studied this question.