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## Homework Statement

Let f(x) = x

^{2}+ 1 in Z

_{3}[x]. Find the order of the quotient ring Z

_{3}[x]/<f>.

## Homework Equations

## The Attempt at a Solution

Note Z

_{3}is a field. Then Z

_{3}[x] is euclidean domain.

Then for any polynomial g(x) can be written as g(x) = p(x).(x

^{2}+1) + r(x) where either r(x) = 0 of deg r(x) < 2.

That is in Z

_{3}[x]/ (x

^{2}+1),

we have g(x) +(x

^{2}+1) = (p(x).(x

^{2}+1) + r(x) )+(x

^{2}+1)

= r(x) + (x

^{2}+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 Z

_{3}[x]/ (x

^{2}+1) are of the form r(x) +(x

^{2}+1). with deg r(x) <2.

So elements are of the form ax+b +(x2+1) in Z3[x]/ (x

^{2}+1), where a, b ranges over the elements of Z

_{3}.

So the number of elements in Z

_{3}[x]/ (x

^{2}+1) is 3x3 = 9.

Hence the order of Z

_{3}[x]/ (x

^{2}+1) = 9.