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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 $a_1, a_2, ... a_k$ be the roots of an irreducible monic polynomial in $Z_p[x]$. One can show that the limit, as $n$ tends to infinity, of $a1^{p^n} + a_2^{p^n} + ... + a_k^{p^n}$ also exists as a $p$-adic integer. Is there a characterization of this $p$-adic integer analogous to the above characterization?

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?

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?

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 $a_1, a_2, ... a_k$ be the roots of an irreducible monic polynomial in $Z_p[x]$. One can show that the limit, as $n$ tends to infinity, of $a1^{p^n} + a_2^{p^n} + ... + a_k^{p^n}$ also exists as a $p$-adic integer. Is there a characterization of this $p$-adic integer analogous to the above characterization?

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} + ... + ak^{p^n} also exists as a p-adic integer. Is there a characterization of this p-adic integer analogous to the above characterization?

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?