What is the Basis of a Quotient Ring?

[BVM]

In my Abstract Algebra course, it was said that if
E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}.<br />
The basis of E over \mathbb{Z}_{3} is equal to [1,\bar{X}].
But this, honestly, doesn't really make sense to me. Why should \bar{X} be in the basis without it containing any other \bar{X}^n? How did they arrive at that exact basis?

 

[micromass]

Can you write \overline{X}^2 in terms of \overline{X} and 1??

 

[BVM]

Thank you for replying.

I've solved the problem. Whereas I previously thought that I couldn't write any \bar{X}^n in terms of 1 and \bar{X} I've since realized that for instance: \bar{X}^2 = -\bar{X}-\bar{2} = 2\bar{X}+\bar{1}.

My initial mistake as to think that in \mathbb{Z}_3 we can't define the negativity resulting in subtracting that polynomal, but obviously you can just add any 3n to it.