On Quadratic Fields Generated by Discriminants of Irreducible Trinomials

A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant ${\mathrm{\Delta }}_n(a,b)$ of the trinomial $f_{n,a,b}(t)=t^n+at+b$, where $n\ge 5$ is a fixed integer. In particular, it is shown that, under the $abc$-conjecture, for every $n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, the quadratic fields $\text{Q}(\sqrt{{\mathrm{\Delta }}_n(a,b)})$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\text{Q}$ and $\mathrm{\mid }{\mathrm{\Delta }}_n(a,b)\mathrm{\mid }\le X$, as $X\to \mathrm{\infty }$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.