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On quadratic fields generated by discriminants of irreducible trinomials. (English) Zbl 1218.11102

Summary: A. Mukhopadhyay, M. R. Murty and K. Srinivas [Proc. Am. Math. Soc. 137, No. 10, 3219–3226 (2009; Zbl 1217.11093)] 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 \(n\geq 5\) is a fixed integer. In particular, it is shown that, under the \( abc\)-conjecture, for every \(n\equiv 1\bmod 4\), 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 \(|\Delta_n(a,b)|\leq X\), as \( X\to \infty\). We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.

MSC:

11R09 Polynomials (irreducibility, etc.)
11R11 Quadratic extensions
11L40 Estimates on character sums
11N36 Applications of sieve methods

Citations:

Zbl 1217.11093
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References:

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