Polynomials and primitive roots in finite fields

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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) ≡ b1 modulo p.

Original languageEnglish (US)
Pages (from-to)499-514
Number of pages16
JournalJournal of Number Theory
Volume13
Issue number4
DOIs
StatePublished - Nov 1981

ASJC Scopus subject areas

  • Algebra and Number Theory

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