Given a number field $K$, I am interested in the distribution of the $L^\infty$ norm of minimal polynomials (over $\mathbb{Z}$) of numbers in $K$. Also, it is interesting to restrict to numbers $\alpha$ with $K=\mathbb{Q} (\alpha)$.
All such polynomials' discriminants are a square factor away from the discriminant of $K$. So that must say something. And it makes it very interesting - the discriminant is a high-degree multivariate polynomial, so the fact that any discriminant can be achieved many times (even if we allow the distance of square factors) amazes me.
To be precise, so no one is annoyed, the question is exactly: what is the asymptotic behavior and error term for $A(x) = |\{ \alpha \in K | \parallel m(\alpha)\parallel_\infty < x\}|$, and $B(x) = |\{ \alpha \in K | K=\mathbb{Q} (\alpha), \parallel m(\alpha)\parallel_\infty < x\}|$, as $x \rightarrow \infty$?
