Examples of DVRs of residue char p and ramification e

Question

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 $e > 2$?

Answer 1

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

Answer 2

Take any polynomial of degree $e$ that is Eisenstein at $p$, adjoin to $\mathbb{Q}_p$ a root of that polynomial and you will get a totally ramified extension of $\mathbb{Q}_p$ of degree $e$. Moreover, all totally ramified extensions of local fields arise in this way, by adjoining a root of an Eisenstein polynomial. So the ring of integers of that extension will be what you are looking for.

Edit: I should have said that all the DVRs you obtain with the above procedure are of the form $\mathbb{Z}_p[\pi]$, where $\pi$ is a root of the Eisenstein polynomial you started with. See e.g. Serre, Local Fields, p 58.

Answer 3

If you fix a finite residue field, then there will be only a finite number of isomorphism classes of local fields with the given absolute ramification index. However, any (perfect) extension of a residue field yields an (unramified) extension with ramification index $1$, so unramified extensions of ${\mathbb Z}_p[p^{\frac{1/e}}]$ will give you an infinite family.