## Abstract

The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds. Theorem. Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p. Theorem. Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b^{1} modulo p.

Original language | English (US) |
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Pages (from-to) | 499-514 |

Number of pages | 16 |

Journal | Journal of Number Theory |

Volume | 13 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1981 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory