Polynomial functions over finite fields are important in computer science and electrical engineering in that they present a mathematical representation of arithmetic circuits. This paper establishes necessary and sufficient conditions for polynomial functions with coefficients in a finite field and naturally restricted degrees to be compatible with given subfields. Most importantly, this is done for the case where the domain and codomain fields have differing cardinalities. These conditions, which are presented for polynomial rings in one and several variables, are developed via a universal permutation that depends only on the cardinalities of the given fields
