Skip to main content
Springer Nature Link
Log in

On irreducible polynomials of small height over finite fields

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

It is shown that, for an arbitrary functionθ(x) → ∞, for almost all prime numbersp of any interval of the form [N−ɛ,N] there exists an irreducible modulop polynomial with coefficients of orderO(θ(p)).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Adleman L. M., Lenstra H. W.: Finding Irreducible Polynomials over Finite Fields. Proc. 18 ACM Symp. Theory Comp 350–355 (1986)

  2. Perelli A., Pintz J., Salerno S.: Bombieri's Theorem in Short Intervals, 2. Invent. Mathem.79, 1–9 (1985)

    Google Scholar 

  3. Prachar K.: Primzahlverteilung. Berlin, Heidelberg, New York: Springer 1957

    Google Scholar 

  4. Shparlinski I. E.: Coefficients of Primitive Polynomials. Matem. Zametki38 (6), 810–815 Russian (1985)

    Google Scholar 

  5. —: On the Distribution Primitive and Irreducible Polynomials Modulo a Prime, Diskretnaja Matem.1 (1), 117–124 (Russian) (1989)

    Google Scholar 

  6. —: Computational and Algorithmic Problems in Finite Fields, Kluwer AP, Dordrecht, 1992

    Google Scholar 

  7. Stephens P. J.: An average result for Artin's conjecture, Mathematika16 (1), 178–188 (1989)

    Google Scholar 

  8. Timofeev N. M.: The Distribution of Arithmetical Functions over Arithmetic Progressions in Short Intervals. Izvestija Acad. Sci. USSR, Ser. Math.51, 341–362 (1987) (Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of MPCE, Macquarie University, 2109, Sydney, NSW, Australia

    I. E. Shparlinski

Authors
  1. I. E. Shparlinski
    View author publications

    You can also search for this author inPubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shparlinski, I.E. On irreducible polynomials of small height over finite fields. AAECC 7, 427–431 (1996). https://doi.org/10.1007/BF01293260

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01293260

Keywords