# How many algebraic integers exist with degree $\leq k$ and bounds on the modulus of all Galois conjugates?

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.

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Both papers count algebraic numbers of bounded height which is slightly different from what you want. (The height of $\alpha$, with conjugates $\alpha_j$ ($1\leq j\leq d$) is the logarithm of the Mahler measure of its minimal polynomial; for an algebraic integer, $h(\alpha)= d^ {-1} \sum \log\max(1,|\alpha_j|)$. However, while Schanuel counts elements of a given number field, Masser and Vaaler count elements of degree $k$ over a given number field. For small $k$, or over $\mathbf Q$, one even has asymptotic expansions but basically nothing more precise seems to be known. –  ACL Nov 26 '10 at 19:02