Counting irreducible skew polynomials $x^{2n}+ax^n+1$

Abstract

We give a counting formula for the coefficients $a\in\F_{p^h}$ for which $x^{2n}+ax^n+1$ is irreducible in $\F_{p^h}[x;σ]$, for arbitrary positive integers $n$ and $h$, where $σ(a)=a^p$ and $p$ is an arbitrary prime power. The proof reduces the problem to quadratic reduced norms, establishes a uniformity statement for every Frobenius generator, and counts the remaining coefficients by Möbius inversion.