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Counting Hopf Galois Structures on Non-Abelian Galois Field Extensions

✍ Scribed by Scott Carnahan; Lindsay Childs

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
93 KB
Volume
218
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let L be a field which is a Galois extension of the field K with Galois w x group G. Greither and Pareigis GP87 showed that for many G there exist K-Hopf algebras H other than the group ring KG which make L into an Ž H-Hopf Galois extension of K or a Galois H *-object in the sense of w x. Chase and Sweedler CS69 . Using Galois descent, they translated the problem of determining the Hopf Galois structures on LrK into one which depends only on the Galois group G. By this translation, they showed, for example, that any Galois extension with non-abelian G admits w x at least one non-classical Hopf Galois structure. Byott By96 further translated the problem to a more amenable group-theoretic problem, and showed that a Galois extension LrK of fields with group G has a unique Hopf Galois structure, namely that by KG, iff n, the order of G, is a Ž . Burnside number, that is, is coprime to n , Euler's phi-function of n. Ž . This implies that G is cyclic of square-free order.

The observation of Greither and Pareigis is the only one in the literature to this point which gives any information on the number of Hopf Galois structures on Galois field extensions for G non-abelian.

The purpose of this paper is to make a start at determining the number of Hopf Galois structures on LrK for some non-abelian Galois groups G.

Before stating our results, we need to describe Byott's counting formula.

Let n be the order of G and let N be an abstract group with cardinality Ž . n. Let , resp. : N ª Perm N be the maps given by sending to left translation by , resp. right translation by the inverse of . The holomorph Ž . Ž . Ž . Ž . of N, Hol N ; Perm N , is the normalizer of N in Perm N : then *Currently a student at the California Institute of Technology.

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