Notes on Speyer’s paper titled “Some Sums over Irreducible Polynomials”
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.