ABELIAN HOPF GALOIS STRUCTURES ON PRIME-POWER GALOIS FIELD EXTENSIONS

作者：Featherstonhaugh S C^*; Caranti A; Childs L N

来源：Transactions of the American Mathematical Society, 2012, 364(7): 3675-3684.

DOI：10.1090/S0002-9947-2012-05503-6

The main theorem of this paper is that if (N, +) is a finite abelian p-group of p-rank m where m + 1 %26lt; p, then every regular abelian subgroup of the holomorph of N is isomorphic to N. The proof utilizes a connection, observed by Caranti, Dalla Volta, and Sala, between regular abelian subgroups of the holomorph of N and nilpotent ring structures on (N, +). Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let L vertical bar K be a Galois extension of fields with abelian Galois group G. If also L vertical bar K is H-Hopf Galois, where the K-Hopf algebra H has associated group N with N as above, then N is isomorphic to G.

• 出版日期2012-7