Shparlinski, Igor E. On quadratic fields generated by discriminants of irreducible trinomials. (English) Zbl 1218.11102 Proc. Am. Math. Soc. 138, No. 1, 125-132 (2010). 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. Cited in 1 Document MSC: 11R09 Polynomials (irreducibility, etc.) 11R11 Quadratic extensions 11L40 Estimates on character sums 11N36 Applications of sieve methods Keywords:irreducible trinomials; quadratic fields; square-sieve; character sums Citations:Zbl 1217.11093 PDFBibTeX XMLCite \textit{I. E. Shparlinski}, Proc. Am. Math. Soc. 138, No. 1, 125--132 (2010; Zbl 1218.11102) Full Text: DOI arXiv References: [1] Stephen D. Cohen, The distribution of polynomials over finite fields, Acta Arith. 17 (1970), 255 – 271. · Zbl 0209.36001 [2] S. D. Cohen, A. Movahhedi, and A. Salinier, Galois groups of trinomials, J. Algebra 222 (1999), no. 2, 561 – 573. · Zbl 0939.12002 · doi:10.1006/jabr.1999.8033 [3] E. Fouvry and N. Katz, A general stratification theorem for exponential sums, and applications, J. Reine Angew. Math. 540 (2001), 115 – 166. · Zbl 0986.11054 · doi:10.1515/crll.2001.082 [4] D. R. Heath-Brown, The square sieve and consecutive square-free numbers, Math. Ann. 266 (1984), no. 3, 251 – 259. · Zbl 0514.10038 · doi:10.1007/BF01475576 [5] Alain Hermez and Alain Salinier, Rational trinomials with the alternating group as Galois group, J. Number Theory 90 (2001), no. 1, 113 – 129. · Zbl 0990.12004 · doi:10.1006/jnth.2001.2653 [6] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. · Zbl 1059.11001 [7] Rudolf Lidl and Harald Niederreiter, Finite fields, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. With a foreword by P. M. Cohn. · Zbl 1139.11053 [8] Florian Luca and Igor E. Shparlinski, On quadratic fields generated by polynomials, Arch. Math. (Basel) 91 (2008), no. 5, 399 – 408. · Zbl 1162.11051 · doi:10.1007/s00013-008-2656-2 [9] A. Mukhopadhyay, M. R. Murty and K. Srinivas, ‘Counting squarefree discriminants of trinomials under \( abc\)’, Proc. Amer. Math. Soc., 137 (2009), 3219-3226. · Zbl 1217.11093 [10] Bernat Plans and Núria Vila, Trinomial extensions of \Bbb Q with ramification conditions, J. Number Theory 105 (2004), no. 2, 387 – 400. · Zbl 1048.11086 · doi:10.1016/j.jnt.2003.11.001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.