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

DOI:
https://doi.org/10.1090/S0002-9939-09-10074-6

Published electronically:
September 4, 2009

MathSciNet review:
2550176

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Abstract | References | Similar Articles | Additional Information

Abstract: A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant of the trinomial , where is a fixed integer. In particular, it is shown that, under the -conjecture, for every , the quadratic fields are pairwise distinct for a positive proportion of such discriminants with integers and such that is irreducible over and , as . 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

Email:
igor@ics.mq.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-09-10074-6

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.