# Generalized Teichmuller representatives

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?

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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.