Implications of a relation on algebraic numbers
If the elementary symmetric functions $e_k$ of $n$ complex variables are all integers, then all of the power sums are integers. Is the converse true?
It seems quite likely that the answer is in the affirmative. When $n=2$, for instance, Newton's identities for $k=1,2$ and $4$ yield in turn that
$e_1,\ 2e_2$ and $2e_2^2$ are all integers, from which
$e_2$ is also an integer. Similar ad hoc considerations, with the help of a computer program, work at least up to $n=12$. For n=3, the integrality of the first 6 power sums suffices, but for n=4 I had to consider the first 16 power sums.