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Some Hopf Galois structures arising from elementary abelian $ p$-groups

Author: Lindsay N. Childs
Journal: Proc. Amer. Math. Soc. 135 (2007), 3453-3460
MSC (2000): Primary 16W30
Published electronically: June 22, 2007
MathSciNet review: 2336557
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Abstract: Let $ p$ be an odd prime, $ G = Z_p^m$, the elementary abelian $ p$-group of rank $ m$, and let $ \Gamma$ be the group of principal units of the ring $ \mathbb{F}_p[x]/(x^{m+1})$. If $ L/K$ is a Galois extension with Galois group $ \Gamma$, then we show that for $ p \ge 5$, the number of Hopf Galois structures on $ L/K$ afforded by $ K$-Hopf algebras with associated group $ G$ is greater than $ p^s$, where $ s = \frac {(m-1)^2}3 - m$.

References [Enhancements On Off] (What's this?)

  • [By96] N. P. Byott, Uniqueness of Hopf Galois structure of a separable field extension, Comm. Algebra 24 (1996), 3217-3228. MR 1402555 (97j:16051a)
  • [Ch00] L. N . Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, Math. Surveys and Monographs, vol. 80, Amer. Math. Soc., 2000. MR 1767499 (2001e:11116)
  • [Ch05] L. N. Childs, Elementary abelian Hopf Galois structures and polynomial formal groups, J. Algebra 283 (2005), 292-316. MR 2102084 (2005g:16073)
  • [Fe06] S. C. Featherstonhaugh, Abelian Hopf Galois structures on Galois field extensions of prime power order, Ph.D. thesis, Univ. at Albany, 2003.
  • [GP87] C. Greither, B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), 239-258. MR 878476 (88i:12006)

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Additional Information

Lindsay N. Childs
Affiliation: Department of Mathematics and Statistics, University at Albany, Albany, New York 12222

Received by editor(s): February 13, 2006
Received by editor(s) in revised form: August 11, 2006
Published electronically: June 22, 2007
Communicated by: Martin Lorenz
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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