Let f(x) = x2 + 1 in Z3[x]. Find the order of the quotient ring Z3[x]/<f>.
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.