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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 $e > 2$?

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3 Answers 3

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


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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.

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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.

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