Finding Order of Quotient Ring in Z3[x]

[hsong9]

Homework Statement

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

Homework Equations

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.

 

[Billy Bob]

Good job, nice solution.

In fact, your quotient ring Z3/<X^2+1> is a field of 9 elements. If you are not able to show this yet, you will be soon.