Prove that in Euclidean ring R the quotient and remainder are always uniquely determined if and only if R is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases: R can also be a field or the null ring.)
E. Fried algebrapolynomialfunctionRing Theorysuperior algebrasuperior algebra unsolved