Finding the order of a quotient field

[Mr Davis 97]

Homework Statement

Find the order of $\mathbb{Z}_3 [x] / \langle x^2 + 2x + 2 \rangle$ and $\mathbb{Z}_3 [x] / \langle x^2 + x + 2 \rangle$

Homework Equations

The Attempt at a Solution

Is there an efficient method for doing this? Is the answer 27 for both? It would seem that both of these consist of elements of the forms $ax^2 + bx + c + \langle x^2 + 2x + 2 \rangle$ or $ax^2 + bx + c + \langle x^2 + x + 2 \rangle$, and there are three choices for the coefficients a, b, c, so $3^3 = 27$

 

[andrewkirk]

It would seem that both of these consist of elements of the forms $ax^2 + bx + c + \langle x^2 + 2x + 2 \rangle$ or $ax^2 + bx + c + \langle x^2 + x + 2 \rangle$

Are you sure? How did you derive this?

 

[Mr Davis 97]

Are you sure? How did you derive this?

Well I am not sure. But my intuition tells me that all higher order polynomials can be written as lower powers by using the $x^2 = -2x -2$ or $x^2 = -x -2$

 

[andrewkirk]

Well I am not sure. But my intuition tells me that all higher order polynomials can be written as lower powers by using the $x^2 = -2x -2$ or $x^2 = -x -2$

What is the maximum possible order of the remainder polynomial one gets from dividing a polynomial by $x^22+2x+2$?