A sum of binomial coefficients modulo prime powers

(The Proof of (12))


Let be a primitive p th root of unity and recall that as ideals in Q. Define to be the sum on the left side of (12) for each j, so that which belongs to the ideal , for . Therefore , belongs to , but as each is a rational integer, it is divisible by where is the smallest multiple of , which is , and (12) follows immediately.