Just before the above-mentioned article hit reddit, I got to wondering if the structure of the Galois field was affected at all by the choice of polynomial you

Just before the above-mentioned article hit reddit, I got to wondering if the structure of the Galois field was affected at all by the choice of polynomial you used as the modulus. Not really -- regardless of a choice of an irreducible polynomial for a modulus, all the fields you'll obtain are isomorphic, meaning that for two fields K, L with 2^n elements there a one-to-one correspondence f: K -> L, such that f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b). This means that while K and L may be different as sets, they have essentially the same structure.