For , the Bernoulli polynomials of order are defined by the generating function . Here is restricted to the range . These polynomials are monic of degree and belong to . The author obtains irreducibility results for them by studying the polynomials generated by the functions . His method is to study, for a fixed prime , the powers of dividing the denominators of the coefficients of ; these -powers are in fact characterized in terms of the -adic expansion of . The results generalize previous ones by N. Kimura and P. J. McCarthy.
A typical result: If is odd and is an integer in the interval , then has an irreducible factor of degree at least iff a given inequality depending on , , and holds true.