An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions

TL;DR

This paper constructs explicit integral bases and normal bases for Carlitz prime power torsion extensions using hyperderivatives of the Anderson-Thakur function, simplifying previous work in the field.

Contribution

It provides a new explicit basis for the ring of integers in Carlitz torsion extensions and constructs a normal basis, advancing understanding of their algebraic structure.

Findings

Explicit basis for ring of integers in Carlitz torsion extensions

Construction of explicit field normal basis

Simplification of previous results by Anglès-Pellarin

Abstract

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 Angl\`es-Pellarin.