A basis for residual polynomials in 𝑛 variables

Kempnert has established the existence of a basis for residual polynomials in one variable with respect to a composite modulus. A residual polynomial modulo m is by definition a polynomial f(x) with integral coefficients which is divisible by m for every integral value of x, and a residual congruence is written f(x) -(mod n). By a basis for a given modulus is meant a finite set of residual polynomials pi(x) which fulfills two requirements: (i) every residual polynomial modulo m is expressible as a sum of products of pi(x) by polynomials in x with integral coefficients; (ii) no member of the set pi(x) can be written identically equal to a sum of products of the remaining members of the set by polynomials in x with integral coefficients. For this work, the following notation is used. The symbol ,u(d) denotes the least positive integer for which d divides ,u!. A special set of divisors of mn is chosen: separate all divisors of m which exceed 1 into groups such that ,u(d) has the same value for all the d's of a group but different values for the d's of different groups; select the largest d of each group and denote this set by di, , d,. Finally, II(y)=x(x-1) (x-y+1); when x is replaced by xi, the product will be designated by 11j(y); 11(1) is interpreted as 1. Employing this notation, Dickson4 gave a brief proof of the theorem due to Kempner?: