Transactions of the American Mathematical Society, Vol. 37, No. 2 (March 1935), 216-225.

Introduction

Kempner 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 integer coefficients which is divisible by *m* for every integral value of x, and a residual congruence is written f(x) = 0 (mod *m*). By a basis for a given modulus is meant a finite set of residual polynomials p_{i}(x) which fulfills two requirements: (i) every residual polynomial modulo *m* is expressible as a sum of products of p_{i}(x) b polynomials in x with integral coefficients; (ii) no member of the set p_{i}(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 μ(d) denotes the least positive integer for which d divides μ!. A special set of divisors of *m* is chosen: separate all divisors of *m* which exceed 1 into groups such that μ(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 d_{1},..., d_{s}. Finally, Π(μ) = x(x–1)...(x–μ+1); when x is replaced by x_{j}, the product will be denoted by Π_{j}(μ); Π(1) is interpreted as 1. Employing this notation, Dickson gave a brief proof of the theorem due to Kempner:

Every residual polynomial f(x) modulomis a sum of products ofmand (m/d_{i})Π(μ(d_{i})) for i = 1, ..., s by polynomials in x with integral coefficients.

In a later paper, Kempner considered the problem for *n* variables. In attempting to apply Dickson's method to the proof of the existence of a basis for residual polynomials in more than one variable, I found that Kempner had omitted from the set p_{i}(x_{1},...,x_{n}) certain residual polynomials in several variables. This was brought to my attention by an example in two variables modulo 12. For this modulus, the d_{1}, ..., d_{s} are d_{1}=12, d_{2}=6, d_{3}=2; the corresponding μ's are μ_{1}=4, μ_{2}=3, μ_{3}=2. Write q_{i} = *m*/d_{i}. The part of the basis containing one variable is composed of

(1) 12, q_{i}Π_{1}(μ_{i}),
q_{i}Π_{2}(μ_{i})
(i = 1,2,3).

Kempner would include in the basis p_{i}(x_{1}, x_{2}) modulo 12 only the seven terms (1). However, the residual polynomial,

must be added since, as is shown below, it is impossible to write the identity

where c, f_{i}, g_{i} are polynomials in x_{1}, x_{2} with integral coefficients. By use of (x_{1}, x_{2}) = (0,0), (2,0), (0,2) we prove the constant terms of c, f_{3}, g_{3} even. The pair (x_{1}, x_{2}) = (2,2) shows the right side of (2) divisible by 24 and the left side equal to 12.