MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $e > 2$?

share|cite|improve this question
up vote 6 down vote accepted

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

share|cite|improve this answer
You are correct. I'm still figuring all this out, but I'm working on a project where it would be nice to have specific examples. Thanks! – Jeremy West Oct 25 '10 at 0:01

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.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.