A set of integer roots of polynomial Inspired by the problem 9 of a team contest (where $2016$ was replaced by $2014$), I would like to know what can be said about the set of integers (positive and negative ones) $S$ knowing the following two rules:


*

*The numbers $0$ and $2016$ are in $S$.

*If an integer $n$ is a root of a non-zero polynomial with coefficients in $S$, then $n\in S$.


Can we describe all the numbers that are in $S$ for this $2016$-case? (and for the $2014$-case?) 

Considering the polynomial
$$2016x+2016$$
we get $-1 \in S$. This allow us to consider the polynomial
$$2016x^{2016}-x^{2015}-...-1$$
and we get $1 \in S$. Now just write $2016$ in base $2$ : $11111100000$, and take the corresponding polynomial
$$x^{10}+x^{9}+x^8+x^7+x^6+x^5-2016$$
and you get $2 \in S$. Now that you can use the coefficient $2$, you can write $2016$ in base $3$ and do the same thing, so $3 \in S$ and for the same reason $4 \in S$. Since $2016$ is not divisible by $5$, you can't do the same because you have a remainder. Notice also that if $a \in S$, then $-a \in S$, since you can take $x+a$. 
You can do the same thing with $2014$, but you can't get $3$ since $2014$ is not divisible by $3$. I'm not sure though if there is another method to get other numbers in $S$.
