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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2016
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1611.09681 |
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Table of Contents:
- Let $\mathfrak{p}$ be a monic irreducible polynomial in $A:=\mathbb{F}_q[θ]$, the ring of polynomials in the indeterminate $θ$ over the finite field $\mathbb{F}_q$, and let $ζ$ be a root of $\mathfrak{p}$ in an algebraic closure of $\mathbb{F}_q(θ)$. For each positive integer $n$, let $λ_n$ be a generator of the $A$-module of Carlitz $\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[ζ,λ_n] \subset K(ζ, λ_n)$ over $A[ζ] \subset K(ζ)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $ω$ 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.