I am interested in writing (in base $p$) elements of the maximal unramified extension $\mathbb{Q}_p^{\mathrm{unr}}$ of $\mathbb{Q}_p$, or (its completion) the field $\mathrm{W}(\mathbb{F}_p^{\mathrm{alg}})[\frac{1}{p}]$ of Witt vectors of the algebraic closure of $\mathbb{F}_p$, or similar fields.

This means choosing a set $S$ of “digits”, meaning a set, containing $0$, of representatives of the residue classes mod $p$ of the (discrete valuation ring of) elements of nonnegative valuation. Then every element of the field has a unique “base $p$” representation as $\sum_{n\in\mathbb{Z}} a_n p^n$ with $a_n\in S$ (and $a_n=0$ for $n\ll 0$).

My question is essentially what “nice” sets of digits can be found, or, if “nice” is too much to ask for, whether there are any “standard” ones.

  • In the case of $\mathbb{Q}_p$, an obvious choice of digits is $\{0,1,2,\ldots,p-1\}$. This is convenient to work with, and very standard, but there is no “obvious” generalization to extensions of $\mathbb{Q}_p$.

  • In the case of $\mathbb{Q}_p^{\mathrm{unr}}$ or its completion, a theoretically elegant choice is the set of Teichmüller representatives (essentially, prime-to-$p$-roots of unity and $0$): this relates nicely to the definition of Witt vectors; but performing any kind of practical computations with this set is almost hopeless: merely computing the $5$-adic number $1+1$ using this representation gives an empirically random sequence of digits (which is probably not computable in quasilinear time).

To avoid the mess of the second example, I certainly want a set $S$ with the property that the base $p$ expansion of $a+b$ and of $a\cdot b$, for all $a,b\in S$, is finite, or at least, eventually constant (and certainly $1$ and $-1$ should also have eventually constant expansions).

This still leaves a lot of room for choice: for example, I think I can take an $\mathbb{F}_p$ basis of the residue field $\mathbb{F}_p^{\mathrm{alg}}$, arbitrarily lift it to elements algebraic over $\mathbb{Q}_p$ whose minimal polynomials have integer coefficients, and define $S$ to be the set of combinations of those things with coefficients in $\{0,\ldots,p-1\}$. But maybe there's something more intelligent to be done, and that is my question.

Or even if there is nothing more intelligent to be done, is there some kind of “standard” choice that has already appeared in the literature? (For example, there is a “standard” choice for representing the elements of $\mathbb{F}_p^{\mathrm{alg}}$, namely using roots of Conway polynomials, so I'm looking for something vaguely similar.)

  • $\begingroup$ One could take the lexicographically first polynomial over $\mathbb Z$ that is irreducible mod $p$, which is equivalent to lifting each coefficient of the Conway polynomial to an element of $\{0,\dots,p-1\}$. $\endgroup$
    – Will Sawin
    May 5 '16 at 14:40
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    $\begingroup$ @WillSawin For a single degree, certainly; but I'm hoping for some kind of compatibility between the choice of digits for the unramified extensions of degrees $m$, $m'$ and $mm'$: the Conway polynomials have some condition of the form, but they don't seem too relevant here. (I mean, I'm sure it's not too hard to find something that works, but maybe there's something a bit smarter than just "working".) $\endgroup$
    – Gro-Tsen
    May 5 '16 at 14:45
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    $\begingroup$ Good luck in your quest. I can’t imagine anything that would turn out better than the good old Witt polynomials, even if they do look like a mess to the eye. At least they’re “understood”. $\endgroup$
    – Lubin
    May 5 '16 at 15:29

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