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On quadratic fields generated by discriminants of irreducible trinomials

Author: Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 138 (2010), 125-132
MSC (2000): Primary 11R11; Secondary 11L40, 11N36, 11R09
Published electronically: September 4, 2009
MathSciNet review: 2550176
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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 $ \vert\Delta_n(a,b)\vert\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

Keywords: Irreducible trinomials, quadratic fields, square-sieve, character sums.
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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