# Basis of a Quotient Ring

1. Jan 7, 2012

### BVM

In my Abstract Algebra course, it was said that if
$$E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}.$$
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?

2. Jan 7, 2012

### micromass

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

3. Jan 8, 2012

### BVM

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 realised 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.