# Finding the order of a quotient field

1. May 2, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Find the order of $\mathbb{Z}_3 [x] / \langle x^2 + 2x + 2 \rangle$ and $\mathbb{Z}_3 [x] / \langle x^2 + x + 2 \rangle$

2. Relevant equations

3. 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 $3^3 = 27$

Last edited: May 2, 2017
2. May 2, 2017

### andrewkirk

Are you sure? How did you derive this?

3. May 2, 2017

### Mr Davis 97

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 $x^2 = -x -2$

4. May 2, 2017

### andrewkirk

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