TITLE: How many algebraic integers exist with degree $\leq k$ and bounds on the modulus of all Galois conjugates?
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: Schanuel, S. On heights in number fields. Bull. Amer. Math. Soc. 70 1964 262–263. 
Masser, David; Vaaler, Jeffrey D., Counting algebraic numbers with large height. II. Trans. Amer. Math. Soc. 359 (2007), no. 1, 427–445
Edit: As per Kevin's request, more context. The first reference is the basic result in the field and the second is the state of the art. The bound can be improved as expected. The second reference also has lots of additional references. Link to abstract and references (and also paper if your institution subscribes):
http://www.ams.org/journals/tran/2007-359-01/S0002-9947-06-04115-8/home.html