Functiones et Approximatio Commentarii Mathematici

Remarks on the distribution of the primitive roots of a prime

Shane Chern

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Abstract

Let $\mathbb{F}_p$ be a finite field of size $p$ where $p$ is an odd prime. Let $f(x)\in\mathbb{F}_p[x]$ be a~polynomial of positive degree $k$ that is not a $d$-th power in $\mathbb{F}_p[x]$ for all $d\mid p-1$. Furthermore, we require that $f(x)$ and $x$ are coprime. The main purpose of this paper is to give an estimate of the number of pairs $(\xi,\xi^\alpha f(\xi))$ such that both $\xi$ and $\xi^\alpha f(\xi)$ are primitive roots of $p$ where $\alpha$ is a given integer. This answers a question of Han and Zhang.

Article information

Source
Funct. Approx. Comment. Math., Volume 57, Number 1 (2017), 39-46.

Dates
First available in Project Euclid: 28 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1490688014

Digital Object Identifier
doi:10.7169/facm/1612

Mathematical Reviews number (MathSciNet)
MR3704224

Zentralblatt MATH identifier
06864162

Subjects
Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 11L40: Estimates on character sums

Keywords
primitive root character sum Weil bound

Citation

Chern, Shane. Remarks on the distribution of the primitive roots of a prime. Funct. Approx. Comment. Math. 57 (2017), no. 1, 39--46. doi:10.7169/facm/1612. https://projecteuclid.org/euclid.facm/1490688014


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References

  • T.M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. xii+338 pp.
  • L. Carlitz, Sets of primitive roots, Compositio Math. 13 (1956), 65–70.
  • D. Han and W. Zhang, On the existence of some special primitive roots mod $p$, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 58(106) (2015), no. 1, 59–66.
  • J. Johnsen, On the distribution of powers in finite fields, J. Reine Angew. Math. 251 (1971), 10–19.
  • M. Szalay, On the distribution of the primitive roots of a prime, J. Number Theory 7 (1975), 184–188.
  • D. Wan, Generators and irreducible polynomials over finite fields, Math. Comp. 66 (1997), no. 219, 1195–1212.
  • A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207.