I am trying to explicitly formulate an algebraic extension of $\mathbb{Z}_p$ of degree $n$, and my guess is that factoring $\mathbb{Z}_p$ by the polynomial $x^n-2$ would serve its job. I know that the polynomial is irreducible in $\mathbb{Z}$ or $\mathbb{Q}$ by the Eisenstein criterion, but am wondering how I would prove the irreducibility of $x^n-2$ in $\mathbb{Z}_p$.
Is my guess correct? Should I instead factor $\bar{\mathbb{Z}}_p$ with $x^{p^n} -x$ or something...?
Edit
I changed the title of my question from "Is $x^n-2$ irreducible over $\mathbb{Z}_p$?" to "constructing an algebraic extension of degree n of $\mathbb{Z}_p$," the finite field of order $p$. I know that $x^n-2$ is not always irreducible in $\mathbb{Z}_p$ now. How can I construct an algebraic extension?