The term function is here used to denote a rational integral function of y with integral coefficients. Employing a fixed integer m and a fixed function P(y), we shall say that two functions are congruent modulis m and P(y) if their difference can be given the form mq(y)+P(y)Q(y); also that f(y) has an inverse f1(y) if f(y)·f1(y) is congruent to unity modulis m, P(y). Then f(y) and f(y)+k(y)P(y) have the same inverse, so that we may restrict attention to functions of degree less than the degree r of P(y). We proceed to prove the
Theorem. If P(y) is of degree r and is irreducible with respect to each prime factor of m, a function R(y) of degree < r has an inverse modulis m and P(y) if and only if the greatest common divisor d of the coefficients of R(y) is prime to m.