Why do we prefer to let the coefficients of our polynomial be members of a field instead of just members of a ring? For instance, e.g. instead of considering polynomials over GF(p) what "nice" properties do we lose by just letting the coefficients be members of $\mathbb{Z}/12\mathbb{Z}$.
One thing I can think of in the case of finite fields GP(p) is that it allows us to construct polynomials with degree up to $p-1$ (using a general residue ring of order d might result in powers higher than $\phi(d)$ being redundant). But this doesn't seem like good enough motivation, since it would seem like many of the things such as factoring etc. can still be done in this context.