Typical primitive polynomials over integer residue rings

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 xs−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.