On quadratic fields generated by discriminants of irreducible trinomials
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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 $|\Delta _n(a,b)|\le X$, as $X\to \infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.References
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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