On quadratic fields generated by discriminants of irreducible trinomials

Igor E. Shparlinski

On quadratic fields generated by discriminants of irreducible trinomials

Igor E. Shparlinski

School of Computing

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4 Citations (Scopus)

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Abstract

A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant Δ_n(a, b) of the trinomial f_n,a,b(t) = tⁿ + 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 ( √ Δ_n(a, b) ) are pairwise distinct for a positive proportion of suchdiscriminants with integers a and b such that f_n,a,b is irreducible over and |Δn(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.

Original language	English
Pages (from-to)	125-132
Number of pages	8
Journal	Proceedings of the American Mathematical Society
Volume	138
Issue number	1
Publication status	Published - Jan 2010

Bibliographical note

Copyright 2009 American Mathematical Society. First published in Proceedings of the American Mathematical Society, Vol. 138, No. 1, pp. 125-132, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-09-10074-6

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abstract = "A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant Δn(a, b) of the trinomial fn,a,b(t) = tn + 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 ( √ Δn(a, b) ) are pairwise distinct for a positive proportion of suchdiscriminants with integers a and b such that fn,a,b is irreducible over and |Δn(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.",

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N2 - A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant Δn(a, b) of the trinomial fn,a,b(t) = tn + 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 ( √ Δn(a, b) ) are pairwise distinct for a positive proportion of suchdiscriminants with integers a and b such that fn,a,b is irreducible over and |Δn(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.

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On quadratic fields generated by discriminants of irreducible trinomials

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