Let be the set of irreducible polynomials over
. Then
. The paper lists certain examples of
below. These are all expanded as geometric series. As one can see only
contribute to the coefficient of
in the sum
. Why don’t the other irreducible polynomials do the same? This is because these are the only two linear polynomials in
. All other polynomials are of higher degree. Moreover, all other irreducible polynomials have the constant term
; otherwise they would be reducible, as
would be a common factor. Hence
would be of the form
, where
. Now divide both the numerator and denominator by
. So we get an expression of the form
. As
for all
, this is a power series expansion in negative powers of
. Also, as
, all such negative exponents will be less than
. This proves that only the polynomials
and
contribute to the coefficient of
in
.
We now try and understand Theorem 1.1 in this paper. Let be the set of monic irreducible polynomials in
. Then
for any
.
A corollary of this is that is in
Proof of corollary: We have rewritten as
, where
. Why can we do that? This is because for any
. Hence, we’re essentially counting the same thing as before. Aren’t we counting each term
times? Also, every irreducible polynomial is of the form
for some
. Now consider the identity
in
. Why is this true? This is because
can be written as
(just multiply and divide
by
).
Now, as if
, and
otherwise. This is because if
, then
is essentially adding
to itself
times. As the characteristic of the field is
, and as
is essentially
, this sum is equal to the inverse of
, which is exactly
. When
, then
. This can be verified independently.