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

Now let a1, a2, ... ak be the roots of an irreducible monic polynomial in Zp[x].  One can show that the limit, as n tends to infinity, of a1^{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?