广义Wieferich素数及单生成多项式
Generalized Wieferich primes and monogenic polynomials
- Shippensburg University（希平斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究广义Wieferich素数在多项式单生成性中的应用，针对一类不依赖Lucas序列同余条件的三项式给出单生成性判据，并推广至任意$N\ge 4$的$N$-项式。
AI中文摘要：
设$b, p\in {\mathbb Z}$，其中$b\ge 2$且$p\ge 3$为素数。若$b^{p-1}\equiv 1 \pmod{p^2}$，则称$p$为基$b$的广义Wieferich素数，或简称为基-$b$ Wieferich素数。当$b=2$时，$p$也简称为Wieferich素数。我们称一个首一多项式$f(x)\in {\mathbb Z}[x]$是单生成的，如果$f(x)$在${\mathbb Q}$上不可约，且$\{1,\theta,\theta^2,\ldots,\theta^{°(f)-1}\}$构成${\mathbb Q}(\theta)$的整数环的一组基，其中$f(\theta)=0$。最近，关于三项式$x^{2n}+bx^n+b$的单生成性给出了充分必要条件，这些条件涉及基-$b$ Wieferich素数同余式，以及涉及某个Lucas序列的同余式。在本文中，我们针对另一类三项式证明了类似结果，该结果不依赖于任何涉及Lucas序列的同余条件。此外，我们将此结果推广到相关的$N$-项式类，对于任意$N\ge 4$。
英文摘要：
Let $b, p\in {\mathbb Z}$ with $b\ge 2$ and $p\ge 3$ a prime. If $b^{p-1}\equiv 1 \pmod{p^2}$, then $p$ is called a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots,θ^{°(f)-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Recently, necessary and sufficient conditions for the monogenicity of the trinomials $x^{2n}+bx^n+b$ were given that included base-$b$ Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of $N$-nomials, for any $N\ge 4$.
