Nilpotent and abelian Hopf–Galois structures on field extensions
Nilpotent and abelian Hopf–Galois structures on field extensions
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场扩展上的幂零和阿贝尔 Hopf-Galois 结构
DOI：
10.1016/j.jalgebra.2013.02.008
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发表时间：
2012
影响因子：
0.9
通讯作者：
N. Byott
中科院分区：
文献类型：
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作者：
N. Byott
设L/K是群为Γ的域的有限Galois扩张.当r是幂零的时，我们证明了L/K上所有幂零Hopf-Galois结构的计数问题可以归结为r的Sylow子群的相应问题。我们用它来枚举所有的幂零（分别）。阿贝尔）Hopf-Galois结构。当Γ是交换的时，给出了L/K上的交换Hopf-Galois结构具有类型Γ的条件.我们还给出了一个关于n的判别准则，使得n次循环扩张上的Hopf-Galois结构具有循环型.
Let L/K be a finite Galois extension of fields with group Γ. When Γ is nilpotent, we show that the problem of enumerating all nilpotent Hopf–Galois structures on L/K can be reduced to the corresponding problem for the Sylow subgroups of Γ. We use this to enumerate all nilpotent (resp. abelian) Hopf–Galois structures on a cyclic extension of arbitrary finite degree. When Γ is abelian, we give conditions under which every abelian Hopf–Galois structure on L/K has type Γ. We also give a criterion on n such that every Hopf–Galois structure on a cyclic extension of degree n has cyclic type.