Functiones et Approximatio Commentarii Mathematici

Remarks on the distribution of the primitive roots of a prime

Shane Chern

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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

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

First available in Project Euclid: 28 March 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

primitive root character sum Weil bound


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.

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