polynomials over finite field with irreducible factors of odd degrees

Question

It is well-known that the number of monic $n$-degree polynomials over a finite field of size $q$ is $q^n$. How many such degree-$n$ polynomials can be completely factored into only irreducible polynomials of odd degree? Does anyone know of any related literature?

Answer 1

I'm not sure what type of answer is desired (estimating the number of such polynomials, or giving a formula for this number, or...). I just want to remark here that one can approach this question via generating functions. Write $O_n$ for the number of monic degree-$n$ polynomials in $\mathbf{F}_q[x]$ which have only odd-degree irreducible factors, and $I_n$ for the number of monic degree-$n$ irreducible polynomials in $\mathbf{F}_q[x]$. Then $O_n$ is the sum, over all partitions $1\le a_1\le a_2\le...$ of $n$ into odd parts, of the product $I_{a_1} \cdot I_{a_2} \cdot ...$. Now write $$ I(t) = \sum_{\substack{i>0 \\ i \text{ odd}}} I_i t^i \quad\text{ and }\quad O(t) = \sum_{i=1}^{\infty} O_i t^i $$ where $t$ is a formal variable. Then the above identity for $O_n$ becomes $$ O(t) = \sum_{k=1}^{\infty} I(t)^k = -1 + \frac{1}{1-I(t)}. $$ This formulation has the advantage of compactly recording the known information. In some sense, since we know the coefficients of $I(t)$, we also know $O(t)$... although whether this helps resolve a specific question about $O(t)$ will depend on the question.