On quadratic fields generated by discriminants of irreducible trinomials

A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant $\Delta_n(a,b)$ 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 $n \equiv 1 \pmod 4$, the quadratic fields $\mathbb{Q}\left(\sqrt{\Delta_n(a,b)}\right)$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\mathbb{Q}$ and $\vert\Delta_n(a,b)\vert\le X$, as $X\to \infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.