Carlitz素数扭拓的整数导数基

A. Maurischat, R. Perkins
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引用次数: 8

摘要

设$\mathfrak{p}$为$A:=\mathbb{F}_q[\theta]$中的一元不可约多项式,设$\zeta$为$\mathbb{F}_q(\theta)$的代数闭包中$\mathfrak{p}$的一个根,在有限域$\mathbb{F}_q$上的不定域$\theta$上的多项式环。对于每个正整数$n$,设$\lambda_n$为Carlitz $\mathfrak{p}^n$ -扭转的$A$ -模块的生成器。给出了在$\mathfrak{p}$根处求值的Anderson-Thakur函数$\omega$的超导数中的单项式组成的整数环$A[\zeta,\lambda_n] \subset K(\zeta, \lambda_n)$ / $A[\zeta] \subset K(\zeta)$的一个基。我们还为这些扩展给出了显式的域正规基。这建立在(在某些地方,简化)Angles-Pellarin的工作之上。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions
Let $\mathfrak{p}$ be a monic irreducible polynomial in $A:=\mathbb{F}_q[\theta]$, the ring of polynomials in the indeterminate $\theta$ over the finite field $\mathbb{F}_q$, and let $\zeta$ be a root of $\mathfrak{p}$ in an algebraic closure of $\mathbb{F}_q(\theta)$. For each positive integer $n$, let $\lambda_n$ be a generator of the $A$-module of Carlitz $\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[\zeta,\lambda_n] \subset K(\zeta, \lambda_n)$ over $A[\zeta] \subset K(\zeta)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $\omega$ evaluated at the roots of $\mathfrak{p}$. We also give an explicit field normal basis for these extensions. This builds on (and in some places, simplifies) the work of Angles-Pellarin.
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