Is $\frac{\mathbb{Z}_3[X]}{(x^2 + x + 2)}$ a field? What is the cardinality?

Question

Is $\frac{\mathbb{Z}_3[X]}{(x^2 + x + 2)}$ a field? What is the cardinality?

I know that the polynomial is irreducible, so that quotient is actually a field. The ring has characteristic 3, so is the cardinality of the field just $3^{2}$?

Answer 1

Yes, it has just $3^2$ elements. In fact, every equivalence class has one and only one element of the form $ax+b$, with $a,b\in\mathbb{Z}_3$. Since there are $3$ choiced for $a$ and $3$ choices for $b$, there are $3^2$ elements.