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 .
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.