On the Recursive Behaviour of the Number of Irreducible Polynomials with Certain Properties over Finite Fields

Let ${\mathbb{F}}_q$ be the field with $q$ elements and of characteristic $p$. For $a\in {\mathbb{F}}_p$ consider the set \begin{equation*} S_a(n)={f\in$\mathbb{F}$_q[x]\mid\operatorname{deg}(f)=n,~f$\text{ irreducible, monic and}$ \operatorname{Tr}(f)=a}. \end{equation*} In a recent paper, Robert Granger proved for $q=2$ and $n\ge 2$ that $\mathrm{\mid }{S}_1(n)\mathrm{\mid }-\mathrm{\mid }{S}_0(n)\mathrm{\mid }=0$ if $2\nmid n$ and $\mathrm{\mid }{S}_1(n)\mathrm{\mid }-\mathrm{\mid }{S}_0(n)\mathrm{\mid }=\mathrm{\mid }{S}_1(n\mathrm{/}2)\mathrm{\mid }$ if $2\mid n$. We will prove a generalization of this result for all finite fields. This is possible due to an observation about the size of certain subsets of monic irreducible polynomials arising in the context of a group action of subgroups of ${\mathrm{PGL}}_2({\mathbb{F}}_q)$ on monic polynomials. Additionally, it enables us to apply these methods to prove two further results that are very similar in nature.