On the reducible quintic complete base polynomials

Abstract

Let B ( x ) = x m + b m − 1 x m − 1 + ⋯ + b 0 ∈ Z [ x ] . If every element in Z [ x ] / ( B ( x ) Z [ x ] ) has a polynomial representative with coefficients in S = { 0 , 1 , 2 , … , | b 0 | − 1 } then B ( x ) is called a complete base polynomial. We prove that if B ( x ) is a completely reducible quintic polynomial with five distinct integer roots less than −1, then B is a complete base polynomial. This is the best possible result regarding the completely reducible polynomials so far. Meanwhile, we provide a Mathematica program for determining whether an input polynomial B ( x ) is a complete base polynomial or not. The program enables us to experiment with various polynomial examples, to decide if the potential result points in the desired direction and to formulate credible conjectures.