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arXiv · 2611.04767

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

Tomoya Hayata·Takahiro Tsushima
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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.

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Tomoya Hayata, Takahiro Tsushima. 2026-11-03. Counting irreducible skew polynomials $x^{2n}+ax^n+1$. https://arxiv.org/abs/2611.04767

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