The precise question is the following:

Question:Can one reasonably bound the number of algebraic integers $\alpha$ of degree at most $k$ - that means there exists a monic integer polynomial $p$ with $\deg(p) \leq k$ and $p(\alpha)=0$ - and such that $p(\beta)=0$ implies $|\beta| \leq n$, i.e. all Galois conjugates of $\alpha$ have a modulus bounded by $n$.

Obviously, the number of relevant polynomials $p(t) = \sum_{i = 0}^k a_{k-i} t^i$ is bounded since $$|a_i| \leq {{k}\choose {i}} \cdot n^i$$ In particular, the number of such $\alpha$ is finite and one obtains a crude upper bound. One can also make a packing argument by observing that the distance between any two such algebraic integers cannot be too small. I am basically asking whether there are better bounds.