An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions

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 Anglès-Pellarin.

17 pages. Corrected typo