# Independence of number fields generated by roots of Littlewood polynomials

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?