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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On quadratic fields generated by discriminants of irreducible trinomials
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by Igor E. Shparlinski PDF
Proc. Amer. Math. Soc. 138 (2010), 125-132 Request permission

Abstract:

A. Mukhopadhyay, M. R. Murty and K. Srinivas 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 $\mathbb {Q}\left (\sqrt {\Delta _n(a,b)}\right )$ 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 $|\Delta _n(a,b)|\le X$, as $X\to \infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.
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Additional Information
  • Igor E. Shparlinski
  • Affiliation: Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia
  • MR Author ID: 192194
  • Email: igor@ics.mq.edu.au
  • Received by editor(s): March 17, 2009
  • Received by editor(s) in revised form: June 2, 2009, and June 8, 2009
  • Published electronically: September 4, 2009
  • Additional Notes: The author was supported in part by ARC Grant DP0556431
  • Communicated by: Wen-Ching Winnie Li
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 125-132
  • MSC (2000): Primary 11R11; Secondary 11L40, 11N36, 11R09
  • DOI: https://doi.org/10.1090/S0002-9939-09-10074-6
  • MathSciNet review: 2550176