Carlitz Modules and Galois Module Structure II
Copyright policy )Carlitz Modules and Galois Module Structure II
[For part I, see J. Number Theory 62, No. 1, 213-219 (1997; Zbl 0867.11079).] Starting with a result of \textit{M. J. Taylor} [J. Reine Angew. Math. 358, 97-103 (1985; Zbl 0582.12008)], several explicit results on the Galois module structure of the ring of integers with respect to a relative extension of abelian number fields were obtained [see e.g. \textit{N. P. Byott} and \textit{G. Lettl}, J. Theor. Nombres Bordx. 8, 125-141 (1996; Zbl 0859.11059) and the literature cited there]. The investigation of corresponding results for function fields started with \textit{R. J. Chapman} [J. Lond. Math. Soc. (2) 44, 250-260 (1991; Zbl 0749.11049)]. In the present paper, cyclotomic function fields, generated with the help of the Carlitz module, are investigated. It is shown that for relative extensions of such fields, the integers (i.e. the elements integral over the ring of polynomials) are free over the associated order if and only if the extension is Carlitz-Kummer.
- Ibaraki University Japan
Carlitz module, additive Galois module structure, Algebra and Number Theory, Arithmetic theory of algebraic function fields, cyclotomic function fields, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Drinfel'd modules; higher-dimensional motives, etc., associated order
Carlitz module, additive Galois module structure, Algebra and Number Theory, Arithmetic theory of algebraic function fields, cyclotomic function fields, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Drinfel'd modules; higher-dimensional motives, etc., associated order
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