# What is the distribution of the $L^\infty$ norm of minimal polynomials of numbers in a number field?

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 $x \rightarrow \infty$?

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Have you tried a (possibly) simple case, like $K={\bf Q}(i)$? –  Gerry Myerson Apr 8 '10 at 23:39
@Gerry: Both $A(x)$ and $B(x)$ are $\sim c_i x^2$ for some constants, and this will be true for any quadratic field, where constant will depend on discriminant, class number, and regulator. But, for a non-quadratic number field, the coefficients of the minimal polynomial of the generic number given in a $\mathbb{Q}$-basis, will be of high degree, and I can't see why the same asymptotic behavior should remain - even though it probably will. –  Dror Speiser Apr 9 '10 at 11:26