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 ∆n(a, b) of the trinomial fn,a,b(t) = t n + at+ b, where n ≥ 5 is a fixed integer. In particular, it is shown that, under the abc-conjecture, for every n ≡ 1 (mod 4), the quadratic fields Q (√ ∆n(a, b) ) are pairwise distinct for a positive proportion of such discriminants with integers a and b such that fn,a,b is irreducible over Q and |∆n(a, b)| ≤ X, as X → ∞. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.