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?