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Zbl pre05344049
Johnston, Henri
Relative Galois module structure of rings of integers of absolutely Abelian number fields. (English)
[J] J. Reine Angew. Math. 620, 85-103 (2008). ISSN 0075-4102; ISSN 1435-5345

Summary: Let $L/K$ be an extension of number fields where $L/\Bbb Q$ is abelian. We define such an extension to be Leopoldt if the ring of integers $\cal O_{L}$ of $L$ is free over the associated order $\cal A_{L/K}$. Furthermore we define an abelian number field $K$ to be Leopoldt if every finite extension $L/K$ with $L/\Bbb Q$ abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan \& Lim, Bley, and Byott \& Lettl culminate in the proof that the $n$-th cyclotomic field $\Bbb Q^{(n)}$ is Leopoldt for every $n$. In this paper, we generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit generators.

MSC 2000:: *11R32 Galois theory for global fields
11R20 Other abelian and metabelian extensions

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