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Counting squarefree discriminants of trinomials under abc


Authors: Anirban Mukhopadhyay, M. Ram Murty and Kotyada Srinivas
Journal: Proc. Amer. Math. Soc. 137 (2009), 3219-3226
MSC (2000): Primary 11R09; Secondary 11C08
DOI: https://doi.org/10.1090/S0002-9939-09-09831-1
Published electronically: June 8, 2009
MathSciNet review: 2515392
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Abstract: For an odd positive integer $ n\ge 5$, assuming the truth of the $ abc$ conjecture, we show that for a positive proportion of pairs $ (a,b)$ of integers the trinomials of the form $ t^n+at+b \ (a,b\in \mathbb{Z})$ are irreducible and their discriminants are squarefree.

References [Enhancements On Off] (What's this?)

  • 1. B. J. Birch and H. P. F. Swinnerton-Dyer: Note on a problem of Chowla, Acta Arith., 5 (1959), 417-423. MR 0113844 (22:4675)
  • 2. N. Bourbaki: Algebra II, Chapter 5, Springer-Verlag, Berlin, 2003. MR 1994218
  • 3. K. Chakraborty and M. Ram Murty, On the number of real quadratic fields with class number divisible by $ 3$, Proc. Amer. Math. Soc., 131 (2003), no. 1, 41-44. MR 1929021 (2003m:11184)
  • 4. S. D. Cohen: The distribution of polynomials over finite fields, Acta Arith., 17 (1970), 255-271. MR 0277501 (43:3234)
  • 5. C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, 1976. MR 0404173 (53:7976)
  • 6. M. Ram Murty: Exponents of class groups of quadratic fields, Topics in Number Theory (University Park, PA, 1997), Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999, 229-239. MR 1691322 (2000b:11123)
  • 7. H. Osada: The Galois groups of the polynomials $ X^n+aX^l+b$, J. Number Theory, 25 (1987), 230-238. MR 873881 (88c:11059)
  • 8. K. Soundararajan: Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc., 61 (2000), no. 2, 681-690. MR 1766097 (2001i:11128)

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Additional Information

Anirban Mukhopadhyay
Affiliation: Institute of Mathematical Sciences, CIT Campus, Tharamani, Chennai 600 113, India
Email: anirban@imsc.res.in

M. Ram Murty
Affiliation: Department of Mathematics and Statistics, Jeffery Hall, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Email: murty@mast.queensu.ca

Kotyada Srinivas
Affiliation: Institute of Mathematical Sciences, CIT Campus, Tharamani, Chennai 600 113, India
Email: srini@imsc.res.in

DOI: https://doi.org/10.1090/S0002-9939-09-09831-1
Keywords: Irreducible polynomials, squarefree discriminants, quadratic extensions
Received by editor(s): August 5, 2008
Published electronically: June 8, 2009
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 American Mathematical Society

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