Further Arithmetical Functions in Finite Fields

Abstract

In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisationwhere P1, …, Pr are distinct irreducible polynomials (i.e. primes).