Let $f(X)$ be a monic and irreducible polynomial of degree $n$ in $\mathbb{Z}[X]$.
What is the nature of the quotient $\mathbb{Z}[X]/ \langle f(X) \rangle$?
Is it a field, an integral domain..
To what is it isomorphic to? (I read that it will be isomorphic to $\mathbb{Z}^n$).
Let $\theta$ be a root of $f$ in $\mathbb C$. Then $\mathbb{Z}[X]/ \langle f(X) \rangle \cong \mathbb{Z}[\theta] \subseteq \mathbb C$ and so is a domain.
However, $\mathbb{Z}[\theta]$ is not a field because $2$ is not invertible. (Try it!)
Because $f$ is monic, $\mathbb{Z}[\theta]$ is the set of all polynomial expressions $g(\theta)$ of degree less than $n$, the degree of $f$. (This follows from polynomial divisions; $f$ being monic is key here.) In particular, $\mathbb{Z}[\theta] \cong \mathbb Z^n$, as additive groups. But not as rings because $\mathbb Z^n$ is not a domain when $n>1$.