[math-fun] galois fields, 2-var polynomials?

Galois fields talk about polynomials in one variable modulo some irreducible polynomial in that variable. When we go up to two variables, what kinds of structures emerge? Do they map onto larger fields somehow, or rings, or what? I'm looking at something like a polynomial that's nth order in x, mth order in y, with coefficients modulo a prime. If you hold one variable or the other constant, you can treat it like a galois field and reduce it modulo some irreducible polynomial in that variable. -- Mike Stay staym@clear.net.nz