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?



Implications of a relation on algebraic numbers
Proved by grobber ([Alexandru Chirvasitu](https://mathoverflow.net/users/9151/alexandru-chirvasitu)) on AoPS: <http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=157732>


**EDIT:** Also, <http://www.artofproblemsolving.com/Forum/viewtopic.php?f=38&t=335001> might contain a solution (posts #4, #7 and #8).


