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There are two great first examples of complete discrete valuation ring with residue field $\mathbb{F}_p = \mathbb{Z}/p$: The $p$-adic integers $\mathbb{Z}_p$, and the ring of formal power series $(\mathbb{Z}/p)[[x]]$. Any complete DVR over $\mathbb{Z}/p$ is a ring structure on left-infinite strings of digits in $\mathbb{Z}/p$. The difference between these two examples is that in the $p$-adic integers, you add the strings of digits with carries. (In any such ring, you can say that a $1$ in the $j$th place times a 1 in the $k$th place is a 1 in the $(j+k)$th place.) At one point I realized that these two examples are not everything: You can also add with carries but move the carry $k$ places to the left instead of one place to the left. The ring that you get can be described as $\mathbb{Z}_p[p^{1/k}]$, or as the $x$-adic completion of $\mathbb{Z}[x]/(x^k - p)$. This sequence has the interesting feature that the terms are made from $\mathbb{Z}_p$ and have characteristic $0$, but the ring structure converges topologically to $(\mathbb{Z}/p)[[x]]$, which has characteristic $p$.

(Edit: Per Mariano's answer, I have in mind a discrete valuation in the old-fashioned sense of taking values in $\mathbb{Z}$, not $\mathbb{Z}^n$.)

I learned from Jonathan Wise in a question on mathoverflow that these examples are still not everything. If $p$ is odd and $\lambda$ is a non-quadratic residue, then the $x$-adic completion of $\mathbb{Z}[x]/(x^2-\lambda p)$ is a different example. You can also call it $\mathbb{Z}_p[\sqrt{\lambda p}]$.

So my question is, is there is a classification or a reasonable moduli space of complete DVRs with residue field $\mathbb{Z}/p$? Or whose residue field is any given finite field? Or if not a classification, an indexed family that includes every example at least once?

I suppose that the question must be related to the Galois theory of $\mathbb{Q}_p$; maybe the best answer would be a relevant sketch of that theory. But part of my interest is in continuous families of DVR structures on the Cantor set of strings of digits in base $p$.

As question authors often say in mathoverflow, I learned stuff from many of the answers and it felt incomplete to only accept one of them. I upvoted several others, though. Following Kevin, the set of pairs $(R,\pi)$, where $R$ is a CDVR and $\pi$ is a uniformizer, is an explicit indexed family that includes every example (of $R$) at least once. The mixed characteristic cases are parametrized by Eisenstein polynomials, and there is only one same-characteristic choice of $R$.

One side issue that was not addressed as much is continuous families of CDVRs. To explain that concern a little better, all CDVRs with finite residue field are homeomorphic (to a Cantor set). For any given finite residue field, there are many homeomorphisms that commute with the valuation. So you could ask what a continuous family of CDVRs can look like, where both the addition and multiplication laws can vary, but the topology and the valuation stay the same. Now that I have been told what the CDVRs are, I suppose that not all that much can happen. It seems key to look at $\nu(p)$, the valuation of the integer element $p$, in a sequence of such structures (say). If $\nu(p) \to \infty$, then the CDVRs have to converge to $k[[x]]$. Otherwise $\nu(p)$ is eventually constant; it eventually equals some $n$. Then it looks like you can convert the convergent sequence of CDVRs to a convergent sequence of Eisenstein polynomials of degree $n$.

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As question authors often say in mathoverflow, I learned stuff from many of the answers and it felt incomplete to only accept one of them. I upvoted several others, though. Following Kevin, the set of pairs $(R,\pi)$, where $R$ is a CDVR and $\pi$ is a uniformizer, is an explicit indexed family that includes every example (of $R$) at least once. The mixed characteristic cases are parametrized by Eisenstein polynomials, and there is only one same-characteristic choice of $R$.

One side issue that was not addressed as much is continuous families of CDVRs. To explain that concern a little better, all CDVRs are homeomorphic (to a Cantor set). For any given finite residue field, there are many homeomorphisms that commute with the valuation. So you could ask what a continuous family of CDVRs can look like, where both the addition and multiplication laws can vary, but the topology and the valuation stay the same. Now that I have been told what the CDVRs are, I suppose that not all that much can happen. It seems key to look at $\nu(p)$, the valuation of the integer element $p$, in a sequence of such structures (say). If $\nu(p) \to \infty$, then the CDVRs have to converge to $k[[x]]$. Otherwise $\nu(p)$ is eventually constant; it eventually equals some $n$. Then it looks like you can convert the convergent sequence of CDVRs to a convergent sequence of Eisenstein polynomials of degree $n$.

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There are two great first examples of complete discrete valuation ring with residue field $\mathbb{F}_p = \mathbb{Z}/p$: The $p$-adic integers $\mathbb{Z}_p$, and the ring of formal power series $(\mathbb{Z}/p)[[x]]$. Any complete DVR over $\mathbb{Z}/p$ is a ring structure on left-infinite strings of digits in $\mathbb{Z}/p$. The difference between these two examples is that in the $p$-adic integers, you add the strings of digits with carries. (In any such ring, you can say that a $1$ in the $j$th place times a 1 in the $k$th place is a 1 in the $(j+k)$th place.) At one point I realized that these two examples are not everything: You can also add with carries but move the carry $k$ places to the left instead of one place to the left. The ring that you get can be described as $\mathbb{Z}_p[p^{1/k}]$, or as the $x$-adic completion of $\mathbb{Z}[x]/(x^k - p)$. This sequence has the interesting feature that the terms are made from $\mathbb{Z}_p$ and have characteristic $0$, but the ring structure converges topologically to $(\mathbb{Z}/p)[[x]]$, which has characteristic $p$.

(Edit: Per Mariano's answer, I have in mind a discrete valuation in the old-fashioned sense of taking values in $\mathbb{Z}$, not $\mathbb{Z}^n$.)

I learned from Jonathan Wise in a question on mathoverflow that these examples are still not everything. If $p$ is odd and $\lambda$ is a non-quadratic residue, then the $x$-adic completion of $\mathbb{Z}[x]/(x^2-\lambda p)$ is a different example. You can also call it $\mathbb{Z}_p[\sqrt{\lambda p}]$.

So my question is, is there is a classification or a reasonable moduli space of complete DVRs with residue field $\mathbb{Z}/p$? Or whose residue field is any given finite field? Or if not a classification, an indexed family that includes every example at least once?

I suppose that the question must be related to the Galois theory of $\mathbb{Q}_p$; maybe the best answer would be a relevant sketch of that theory. But part of my interest is in continuous families of DVR structures on the Cantor set of strings of digits in base $p$.

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