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Sums of Binomial Coefficients


Proof of (1.12): \ 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 (1.12) for each j, so that

which belongs to the ideal , for . Therefore , belongs to . However, since each is a rational integer, it must be divisible by where is the smallest multiple of , which is , and (1.12) follows immediately.
Proof of (1.13): \ Let d be a quadratic non--residue and a, b and n any positive integers. Define the sequence of integers by

so that, from the binomial theorem,

as , where is the sum in (1.13).

Now, by Kummer's Theorem,

and so, if p+1 divides n then p divides , which divides , by (7.1). So by selecting a=b=1 and letting d run through all quadratic non--residues , we have equations in the unknowns . Therefore each must be divisible by p as these equations give rise to a Vandermonde matrix whose determinant is not divisible by p.

On the other hand if (1.13) holds for all odd j then is divisible by p for any admissible choices of a,b and d, by (7.2). Now fix d and select a and b so that is a primitive root modulo p in the field Q. Note that , so that . By (7.1), we see that , and so . But is a primitive root modulo p and so divides , giving that p+1 divides n.