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 such generalization transfers acrossmathematically equivalent representations of the same operation. Westudy this question through multiplication in finite fields underchanges of basis, where the Galois action defines a natural equivalenceamong basis representations. This setting permits controlled trainingand test splits that separate learning an algebraic operation fromtransferring it to unseen basis representations. We compare severalways of providing or recovering the relevant structure, includingexplicit invariant information, basis matrices, orbit recognition,algebraic decomposition, and canonicalization constructed from alearned Galois action. We find that transfer depends strongly on howthe equivalence structure is presented to the model. Mathematicallyrelevant information does not necessarily provide a learning advantage,while relational and constructive approaches can recover structurethat direct identification does not. These results show that learningan operation and transferring it across equivalent representations aredistinct problems, and provide a controlled framework for studying howalgebraic symmetry can support invariant generalization.