Polynomials and primitive roots in finite fields |
| |
Authors: | Daniel J Madden |
| |
Institution: | Department of Mathematics, University of Arizona, Tucson, Arizona 85721 USA |
| |
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. |
| |
Keywords: | |
本文献已被 ScienceDirect 等数据库收录! |
|