Abstract:A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod{p^2}$.
A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and
$\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$
is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we show that ${\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime.
| Subjects: | Number Theory (math.NT) |
| Cite as: | arXiv:2605.13460 [math.NT] |
| (or arXiv:2605.13460v2 [math.NT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.13460 arXiv-issued DOI via DataCite |
From: Lenny Jones Ph.D. [view email]
[v1]
Wed, 13 May 2026 12:52:46 UTC (10 KB)
[v2]
Fri, 22 May 2026 10:35:24 UTC (10 KB)
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