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Taming wild extensions with Hopf algebras. (English) Zbl 0632.12013

At the beginning of this paper is Noether’s theorem: For a Galois extension \(L\supset K\) with Galois group G and rings of integers \(S\supset R\), respectively, S is locally RG-isomorphic to RG iff L/K is tamely ramified. The local isomorphisms between RG and S mean the existence of a normal basis in S. Since these isomorphisms fail in the non-tame case one has tried to replace RG by another order \({\mathfrak A}\) in KG. Generalizations of Noether’s theorem have been found for \(K={\mathbb{Q}}\), the rational numbers. But for general number fields even this approach failed.
The author now investigates those orders \({\mathfrak A}\) of S in KG which are R-Hopf algebras with the structure induced by KG. He shows that for an abelian extension \(L\supset K\) of number fields with group G and rings of integers \(R\subset S\) and \({\mathfrak A}\), the order of S in KG, a Hopf algebra, S is locally \({\mathfrak A}\)-isomorphic to \({\mathfrak A}\) and S a tame \({\mathfrak A}\)-extension of R. So the Hopf algebra property of \({\mathfrak A}\) is most important, but more: a wild RG-extension S becomes a tame \({\mathfrak A}\)-extension (with suitable definition of tameness). The author then investigates conditions for cyclotomic extensions of \({\mathbb{Q}}\) and Kummer extensions L/K of prime order, such that \({\mathfrak A}\) is a Hopf algebra. For this purpose he makes intensive use of the Tate-Oort classification of group schemes of order p over rings of integers.
Reviewer: B.Pareigis

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
13B05 Galois theory and commutative ring extensions
11R18 Cyclotomic extensions
11S15 Ramification and extension theory
14E22 Ramification problems in algebraic geometry
14L15 Group schemes
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