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

Abstract

Neural networks can learn algebraic operations from finite examples,but it remains unclear whether this ability transfers acrossmathematically equivalent representations of the same operation. Westudy this question through multiplication in finite fields underchanges of basis. The Galois action organizes basis representationsinto orbits, and bases in the same orbit induce the same coordinatemultiplication map. This structure allows us to separate learningmultiplication from transferring it to basis representations that arenot used for training. We examine several ways of providing orrecovering the relevant orbit structure, including invariant labels,basis matrices, orbit recognition, and algebraic decomposition. Ourmain approach trains a model to predict the Galois action between basisrepresentations. Repeated applications of the learned transformationare then used to construct a canonical representative for each orbit,which supports multiplication on held-out bases through exactcanonical matching. This provides a concrete mechanism for convertinga learned algebraic symmetry into an invariant representation that canbe used for transfer.