Fix a prime $p$. The Teichmuller representative associated to a $p$-adic integer $a$ is the unique root of $x^p - x$ in $Z\_p$ congruent to $a$ mod $p$. One can identify this representative with the limit, as $n$ tends to infinity, of $a^{p^n}$.


Now let $a\_1, a\_2, \ldots, a\_k$ be the roots of an irreducible monic polynomial in $Z\_p[x]$. One can show that the limit, as $n$ tends to infinity, of $a\_1^{p^n} + a\_2^{p^n} + ... + a\_k^{p^n}$ also exists as a $p$-adic integer. Is there a characterization of this $p$-adic integer analogous to the above characterization?



Generalized Teichmuller representatives
I think I can give a characterization of your limit as a sum of Teichmüller representatives.


Let $q = p^f$ be some power of $p$. Let $Z\_q = W(F\_q)$ be the valuation ring of the unramified extension of $Q\_p$ of degree $f$. Then for any $a$ in $Z\_q$, there is a unique root of $x^q - x$ in $Z\_q$ congruent to $a$ mod $p$. One can identify this with the limit, as n tends to infinity, of $a^{q^n}$.


I've never seen this before, but I guess you can do the same thing even if your extension is ramified. Let $R$ be some finite extension of $Z\_p$. Let $F\_q$ denote its residue field. Then for any $a$ in $R$, there is a unique root of $x^q - x$ in $R$ congruent to $a$ mod $p^{1/e}$, where $e$ is the ramification index. Again, it can be identified with the limit of $a^{q^n}$. 


Assuming the limit you mentioned exists, it is the same as the limit of $a\_1^{q^n} + \cdots + a\_k^{q^n}$. And then this limit is the sum of the Teichmüller representatives that I just described.


