Abstract Algebra: $\mathbb{Z}[X]$, polynomial ring

Question

In abstract algebra, we're currently going over the topic of Polynomial Rings. I understand that $$\mathbb{Z}[X]=a_0+a_1x^1+a_2x^2+...+a_nx^n$$ but I get confused to how exactly $\mathbb{Z}_3[X]$ and $\mathbb{Z}_5[X]$ is suppose to look like.

Answer 1

For ${\bf{Z}}_{3}[X]$, the elements are $a_{0}+a_{1}x+\cdots+a_{n}x^{n}$ with $a_{0},a_{1},...,a_{n}\in{\bf{Z}}_{3}$.

Answer 2

$\mathbb{Z}_p[T]$ refers to a polynomial rings over a finite fields $p$, with others words:

$$\mathbb{Z}_p[T] := \sum_{i = 0}^n (a_iT^i \text{ mod }p)$$