On Quadratic Fields Generated by Discriminants of Irreducible Trinomials

TL;DR

This paper investigates the quadratic fields generated by discriminants of certain irreducible trinomials, providing both conditional and unconditional results on their distinctness and distribution.

Contribution

It extends previous work by establishing an unconditional result on the pairwise distinctness of quadratic fields from discriminants of irreducible trinomials, using the square-sieve and character sum bounds.

Findings

Conditional result assuming abc-conjecture for n ≡ 1 mod 4

Unconditional weaker version proved using sieve methods

Demonstrates abundance of distinct quadratic fields from discriminants

Abstract

A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) 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 $\Q(\sqrt{\Delta_n(a,b)})$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\Q$ and $|\Delta_n(a,b)|\le X$, as $X\to \infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.