Polynomials and primitive roots in finite fields

Abstract

Listen

Translate article

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.