Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^k \neq \beta^l}}' \frac{[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]}{[\mathbb{Q}(\alpha):\mathbb{Q}] \cdot [\mathbb{Q}(\beta):\mathbb{Q}]} \leq 1, $$ where the minimum is over all pairs of multiplicatively independent $\alpha, \beta \in \mathcal{R}_d$. This theorem of Amoroso and David implies the lower bound $c(d) \gg (\log{d})^{-20}$. Thus, in particular, it follows easily that there are only finitely many irreducible and non-cyclotomic $\{-1,0,1\}$-polynomials $f(X)$ for which the number field $\mathbb{Q}[X] / (f(X))$ is Galois over $\mathbb{Q}$.

I wondered how small an upper bound on $c(d)$, better than the trivial $c(d) \leq 1$, may one prove or expect to have. In particular: is $c(d)$ actually bounded from below by an absolute positive constant, or goes to $0$ as $d$ increases?


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.