On quadratic fields generated by discriminants of irreducible trinomials
Author:
Igor E. Shparlinski
Journal:
Proc. Amer. Math. Soc. 138 (2010), 125132
MSC (2000):
Primary 11R11; Secondary 11L40, 11N36, 11R09
Published electronically:
September 4, 2009
MathSciNet review:
2550176
Fulltext PDF Free Access
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 squaresieve and bounds of character sums to obtain a weaker but unconditional version of this result.
 1.
Stephen
D. Cohen, The distribution of polynomials over finite fields,
Acta Arith. 17 (1970), 255–271. MR 0277501
(43 #3234)
 2.
S.
D. Cohen, A.
Movahhedi, and A.
Salinier, Galois groups of trinomials, J. Algebra
222 (1999), no. 2, 561–573. MR 1734229
(2001b:12004), 10.1006/jabr.1999.8033
 3.
E.
Fouvry and N.
Katz, A general stratification theorem for exponential sums, and
applications, J. Reine Angew. Math. 540 (2001),
115–166. MR 1868601
(2003e:11088), 10.1515/crll.2001.082
 4.
D.
R. HeathBrown, The square sieve and consecutive squarefree
numbers, Math. Ann. 266 (1984), no. 3,
251–259. MR
730168 (85h:11050), 10.1007/BF01475576
 5.
Alain
Hermez and Alain
Salinier, Rational trinomials with the alternating group as Galois
group, J. Number Theory 90 (2001), no. 1,
113–129. MR 1850876
(2002f:12004), 10.1006/jnth.2001.2653
 6.
Henryk
Iwaniec and Emmanuel
Kowalski, Analytic number theory, American Mathematical
Society Colloquium Publications, vol. 53, American Mathematical
Society, Providence, RI, 2004. MR 2061214
(2005h:11005)
 7.
Rudolf
Lidl and Harald
Niederreiter, Finite fields, 2nd ed., Encyclopedia of
Mathematics and its Applications, vol. 20, Cambridge University Press,
Cambridge, 1997. With a foreword by P. M. Cohn. MR 1429394
(97i:11115)
 8.
Florian
Luca and Igor
E. Shparlinski, On quadratic fields generated by polynomials,
Arch. Math. (Basel) 91 (2008), no. 5, 399–408.
MR
2461203 (2010c:11119), 10.1007/s0001300826562
 9.
A. Mukhopadhyay, M. R. Murty and K. Srinivas, `Counting squarefree discriminants of trinomials under ', Proc. Amer. Math. Soc., 137 (2009), 32193226.
 10.
Bernat
Plans and Núria
Vila, Trinomial extensions of ℚ with ramification
conditions, J. Number Theory 105 (2004), no. 2,
387–400. MR 2040165
(2005a:11176), 10.1016/j.jnt.2003.11.001
 1.
 S. D. Cohen, `The distribution of polynomials over finite fields', Acta Arith., 17 (1970), 255271. MR 0277501 (43:3234)
 2.
 S. D. Cohen, A. Movahhedi and A. Salinier, `Galois groups of trinomials', J. Algebra, 222 (1999), 561573. MR 1734229 (2001b:12004)
 3.
 E. Fouvry and N. Katz, `A general stratification theorem for exponential sums, and applications', J. Reine Angew. Math., 540 (2001), 115166. MR 1868601 (2003e:11088)
 4.
 D. R. HeathBrown, `The square sieve and consecutive squarefree numbers', Math. Ann., 266 (1984), 251259. MR 730168 (85h:11050)
 5.
 A. Hermez and A. Salinier, `Rational trinomials with the alternating group as Galois group', J. Number Theory, 90 (2001), 113129. MR 1850876 (2002f:12004)
 6.
 H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc., Providence, RI, 2004. MR 2061214 (2005h:11005)
 7.
 R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, Cambridge, 1997. MR 1429394 (97i:11115)
 8.
 F. Luca and I. E. Shparlinski, `Quadratic fields generated by polynomials', Arch. Math. (Basel), 91 (2008), 399408. MR 2461203
 9.
 A. Mukhopadhyay, M. R. Murty and K. Srinivas, `Counting squarefree discriminants of trinomials under ', Proc. Amer. Math. Soc., 137 (2009), 32193226.
 10.
 B. Plans and N. Vila, `Trinomial extensions of with ramification conditions', J. Number Theory, 105 (2004), 387400. MR 2040165 (2005a:11176)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11R11,
11L40,
11N36,
11R09
Retrieve articles in all journals
with MSC (2000):
11R11,
11L40,
11N36,
11R09
Additional Information
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia
Email:
igor@ics.mq.edu.au
DOI:
http://dx.doi.org/10.1090/S0002993909100746
Keywords:
Irreducible trinomials,
quadratic fields,
squaresieve,
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:
WenChing Winnie Li
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
