TY - JOUR

T1 - Polynomials and primitive roots in finite fields

AU - Madden, Daniel J.

PY - 1981/11

Y1 - 1981/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0040449310&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040449310&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(81)90041-X

DO - 10.1016/0022-314X(81)90041-X

M3 - Article

AN - SCOPUS:0040449310

VL - 13

SP - 499

EP - 514

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 4

ER -