1. The problem statement, all variables and given/known data Let f(x) = x2 + 1 in Z3[x]. Find the order of the quotient ring Z3[x]/<f>. 2. Relevant equations 3. The attempt at a solution Note Z3 is a field. Then Z3[x] is euclidean domain. Then for any polynomial g(x) can be written as g(x) = p(x).(x2+1) + r(x) where either r(x) = 0 of deg r(x) < 2. That is in Z3[x]/ (x2+1), we have g(x) +(x2+1) = (p(x).(x2+1) + r(x) )+(x2+1) = r(x) + (x2+1) That is every polynomial is equalent to either zero polynomial or a polynomial of degree less than 2. So we have the elements in Z3[x]/ (x2+1) are of the form r(x) +(x2+1). with deg r(x) <2. So elements are of the form ax+b +(x2+1) in Z3[x]/ (x2+1), where a, b ranges over the elements of Z3. So the number of elements in Z3[x]/ (x2+1) is 3x3 = 9. Hence the order of Z3[x]/ (x2+1) = 9.