Typical primitive polynomials over integer residue rings

Abstract

Let N be a product of distinct prime numbers and Z / ( N ) be the integer residue ring modulo N. In this paper, a primitive polynomial f ( x ) over Z / ( N ) such that f ( x ) divides x s − c for some positive integer s and some primitive element c in Z / ( N ) is called a typical primitive polynomial. Recently typical primitive polynomials over Z / ( N ) were shown to be very useful, but the existence of typical primitive polynomials has not been fully studied. In this paper, for any integer m ⩾ 1 , a necessary and sufficient condition for the existence of typical primitive polynomials of degree m over Z / ( N ) is proved.