From Symmetry to Invariance: Learning Galois Equivalent Representations in Finite Fields

Neural networks can learn algebraic operations from finite examples, but it remains unclear whether this ability transfers across mathematically equivalent representations of the same operation. We study this question through multiplication in finite fields under changes of basis. The Galois action organizes basis representations into orbits, and bases in the same orbit induce the same coordinate multiplication map. This structure allows us to separate learning multiplication from transferring it to basis representations that are not used for training. We examine several ways of providing or recovering the relevant orbit structure, including invariant labels, basis matrices, orbit recognition, and algebraic decomposition. Our main approach trains a model to predict the Galois action between basis representations. Repeated applications of the learned transformation are then used to construct a canonical representative for each orbit, which supports multiplication on held-out bases through exact canonical matching. This provides a concrete mechanism for converting a learned algebraic symmetry into an invariant representation that can be used for transfer.