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Primitive values of quadratic polynomials in a finite field

Andrew R. Booker, Stephen D. Cohen, Nicole Sutherland, Tim Trudgian

TL;DR

It is proved that for all $q>211$, there always exists a primitive root $g$ in the finite field $\mathbb{F}_{q}$ such that Q(g)$ is also a primitiveRoot.

Abstract

We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field $\mathbb{F}_{q}$ such that $Q(g)$ is also a primitive root, where $Q(x)= ax^2 + bx + c$ is a quadratic polynomial with $a, b, c\in \mathbb{F}_{q}$ such that $b^{2} - 4ac \neq 0$.

Primitive values of quadratic polynomials in a finite field

TL;DR

It is proved that for all , there always exists a primitive root in the finite field such that Q(g)$ is also a primitiveRoot.

Abstract

We prove that for all , there always exists a primitive root in the finite field such that is also a primitive root, where is a quadratic polynomial with such that .

Paper Structure

This paper contains 7 sections, 9 theorems, 44 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A cast of character sums
  3. Introducing the sieve
  4. The modified prime sieve
  5. Computation
  6. Small values of $q$
  7. A further algorithm for odd $q$

Key Result

Theorem 1

Let $a,b,c \in \mathbb{F}_{q}$ with $a\neq0$ and with $b^{2} - 4ac \neq 0$. Then there are primitive roots $g_{n}$ and $g_{m}$ such that for all $q$ with the exception of the values listed in $(exceptions)$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • Lemma 3: Lemma 1 COT
  • Lemma 4
  • proof
  • Theorem 3
  • ...and 5 more