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 \(p_i(x)\) 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 \(\mu(d)\) denotes the least positive integer for which *d* divides \(\mu!\). A special set of divisors of *m* is chosen: separate all divisors of *m* which exceed 1 into groups such that \(\mu(d)\) 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, \(\Pi(\mu) = x(x-1)\dots(x-\mu+1);\) when *x* is replaced by *x*_{j}, the product will be denoted by \(\Pi_j(\mu)\); \(\Pi(1)\) 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 \((m/d_i)\Pi(\mu(d_i))\) for \(i=1,\dots,s\) by polynomials inxwith 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 \(p_i(x_1,\dots,x_n)\) certain residual polynomials in several variables. This was brought to my attention by an example in two variables modulo 12. For this modulus, the \(d_1,\dots,d_s\) are \(d_1=12\), \(d_2=6\), \(d_3=2\); the corresponding \(\mu\)'s are \(\mu_1=4\), \(\mu_2=3\), \(\mu_3=2\). Write \(q_i=m/d_i\). The part of the basis containing one variable is composed of

(1) 12, \(q_i\Pi_1(\mu_i)\), \(q_i\Pi_2(\mu_i)\) (*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,\[P = (m/(d_3 \cdot d_3))\Pi_1(\mu_3)\Pi_2(\mu_3) = 3x_1(x_1-1)x_2(x_2-1)\]must be added since, as is shown below, it is impossible to write the identity

(2) \(\displaystyle P=12\cdot c + \sum_{i=1}^3q_i\Pi_1(\mu_i)f_i + \sum_{i=1}^3q_i\Pi_2(\mu_i)g_i\)

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.