Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold

2009/08/31 by Nigel P. Byott, Byott, Nigel P., G. Griffith Elder +1
Computer Science · Mathematics · #11S23 #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT #msc:11S23 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0908.4562

arxiv created 2009/08/31 · openalex publication_date 2009/08/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let k be a perfect field of characteristic p and let K=k((T)). For the class of characteristic p elementary abelian p-extensions L/K with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring \mathfrakO_L to be free over its associated order A_L/K in K[\Gal(L/K)]. Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.

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