Archiv der Mathematik

, Volume 91, Issue 5, pp 399–408

On quadratic fields generated by polynomials

Article

DOI: 10.1007/s00013-008-2656-2

Cite this article as:
Luca, F. & Shparlinski, I.E. Arch. Math. (2008) 91: 399. doi:10.1007/s00013-008-2656-2

Abstract.

Let \(f(X) \in {\mathbb{Z}}[X]\) be a polynomial of degree d ≥ 2 without multiple roots. Under the assumption of the ABC-conjecture, an asymptotic formula for the number of distinct fields among \({\mathbb{Q}}\left({\sqrt {f(n)}}\right)\) for \(n \in \{1, \ldots, N\}\) has recently been given by Cutter, Granville, and Tucker. We use bounds for character sums to obtain an unconditional lower bound on the number of such fields for \(n \in \{M + 1, \ldots, M + N\}\).

Keywords.

Quadratic fields square sieve character sums 

Mathematics Subject Classification (2000).

Primary 11R11 Secondary 11L40, 11N36 

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autonoma de MéxicoMoreliaMéxico
  2. 2.Department of ComputingMacquarie UniversitySydneyAustralia

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