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** 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**.

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** Next:** Bernoulli numbers and
**Up:** Arithmetic Properties of Binomial
** Previous:** Sums of Binomial