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 $e > 2$?



Examples of DVRs of residue char p and ramification e
The family of rings $\mathbb{Z}\_p[p^{\frac{1}{e}}]$ does what you want.


Because of your question, I gather you do not yet know the correspondence between totally ramified extensions and Eisenstein polynomials. For this see e.g. Serre's *Local Fields*, Lang's *Algebraic Number Theory*, or Section 4.3 of


[http://alpha.math.uga.edu/~pete/8410Chapter4.pdf](http://alpha.math.uga.edu/%7Epete/8410Chapter4.pdf)


