Classification of the Hopf Galois Structures on Prime Power Radical Extensions
Classification of the Hopf Galois Structures on Prime Power Radical Extensions
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素幂根式扩张的Hopf伽罗瓦结构的分类
DOI：
10.1006/jabr.1998.7479
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发表时间：
1998
影响因子：
0.9
通讯作者：
Timothy Kohl
中科院分区：
文献类型：
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作者：
Timothy Kohl
设p是奇素数，n是正整数，k是特征为零的域。设K = k（w），其中w pn = a ∈ k，其中a使得[ K：k ] = pn，r表示0到n之间的最大整数，使得K ∈ k（n p r）= k（n p r），其中n p r表示本原p r次单位根.扩展K / k是可分的，但不一定正常，并通过Greither和Pareigis，是H -伽罗瓦与H一个K -霍普夫代数形式的群环kN其中K是正常的封闭K / k。如果N（K / k），则称H几乎是经典的。其结果是，如果r n，那么有p r的Hopf伽罗瓦结构的K / k，其中相关的群N是循环的顺序p n。其中，p min（r，n-r）是几乎经典的，其余的是非几乎经典的。当r = n时，存在p n − 1个H伽罗瓦结构，N <$C p n，其中只有一个几乎是经典的。最后，我们表明，这些是唯一可能的结构。也就是说，对于这类扩张，N必须是循环的。
Abstract Let p be an odd prime and n a positive integer and let k be a field of characteristic zero. Let K = k ( w ) with w p n = a ∈ k where a is such that [ K : k ] = p n and let r denote the largest integer between 0 and n such that K ∩ k (ζ p r ) = k (ζ p r ), where ζ p r denotes a primitive p r th root of unity. The extension K / k is separable, but not necessarily normal and, by Greither and Pareigis, is H -Galois with H a K -Hopf algebra form of a group ring kN where K is the normal closure of K / k . H is said to be almost classical if N K / k ). The result is that if r n then there are p r Hopf Galois structures on K / k for which the associated group N is cyclic of order p n . Of these, p min ( r , n − r ) are almost classical and the rest are non-almost classical. When r = n , there are p n − 1 H -Galois structures for which N ≅ C p n of which only one is almost classical. Finally, we show that these are the only structures possible. That is, for this class of extensions, N must be cyclic.