TY - JOUR

T1 - On quadratic fields generated by discriminants of irreducible trinomials

AU - Shparlinski, Igor E.

N1 - 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

PY - 2010/1

Y1 - 2010/1

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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77951452696&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77951452696

SN - 0002-9939

VL - 138

SP - 125

EP - 132

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -