# Implications of a relation on algebraic numbers

Assume that $\alpha_1,\ldots,\alpha_n$ are algebraic numbers. Assuming that

$\sum_{i=1}^n \alpha_i^k \in \mathbb{Z}$

for all $k\in\mathbb{N}$. Does this imply that the $\alpha_i$ are actually algebraic integers? I know that if these $\alpha_i$ are the conjugates of some algebraic number $\alpha$, then the relation implies that $\textrm{Tr}(\alpha^k)\in\mathbb{Z}$ for all $k\in\mathbb{N}$ (trace taken over e.g. $\mathbb{Q}(\alpha)/\mathbb{Q}$). This implies that $\alpha$ is an algebraic integer, so in this special case it's true.

Does anyone know about the general case?

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vess claimed to have proven something slightly better here: artofproblemsolving.com/Forum/viewtopic.php?p=178197#p178197 –  darij grinberg Sep 3 '11 at 22:55