# A sum of binomial coefficients modulo prime powers

## (The Proof of (12))

**(12)**

* Proof:*

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.