I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the characteristic of the field obtained by the quotient of $R$ with its unique maximal ideal $M$ and by ramification index I mean the largest positive integer $e$ such that $M^e \supseteq pR$.

Without the restriction on the ramification, a simple example is the $p$-adic integers $\mathbb{Z}_p$. However, when we try to fix the ramification index, this becomes more challenging. For example, with $e = 2$ we can take $R = \mathbb{Z}_p[\sqrt{p}]$. The maximal ideal of this ring is $M = \sqrt{p}R$ which has ramification index 2.

My question: is there a simple construction for such a ring with arbitrary $p$ and $e$? If not, can an infinite family of such rings be constructed that have a known ramification index $e > 2$?