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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Taming wild extensions with Hopf algebras


Author: Lindsay N. Childs
Journal: Trans. Amer. Math. Soc. 304 (1987), 111-140
MSC: Primary 11R33; Secondary 16A24
MathSciNet review: 906809
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Abstract: Let $ K \subset L$ be a Galois extension of number fields with abelian Galois group $ G$ and rings of integers $ R \subset S$, and let $ \mathcal{A}$ be the order of $ S$ in $ KG$. If $ \mathcal{A}$ is a Hopf $ R$-algebra with operations induced from $ KG$, then $ S$ is locally isomorphic to $ \mathcal{A}$ as $ \mathcal{A}$-module. Criteria are found for $ \mathcal{A}$ to be a Hopf algebra when $ K = {\mathbf{Q}}$ or when $ L/K$ is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over $ R$ in $ L$ which are tame or Galois $ H$-extensions, $ H$ a Hopf order in $ KG$, using a generalization of the discriminant.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0906809-8
Article copyright: © Copyright 1987 American Mathematical Society