Fixedpoint free endomorphisms and Hopf Galois structures
Author:
Lindsay N. Childs
Journal:
Proc. Amer. Math. Soc. 141 (2013), 12551265
MSC (2000):
Primary 12F10
Published electronically:
September 21, 2012
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Abstract: Let be a Galois extension of fields with finite Galois group . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on and regular subgroups of normalized by , and Byott translated the problem into that of finding equivalence classes of embeddings of in the holomorph of groups of the same cardinality as . In 2007 we showed, using Byott's translation, that fixed point free endomorphisms of yield Hopf Galois structures on . Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the GreitherPareigis approach and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are ``twistings'' of the Hopf Galois structure by , the Hopf algebra that arises from the left regular representation of in . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.
 [By96]
N.
P. Byott, Uniqueness of Hopf Galois structure for separable field
extensions, Comm. Algebra 24 (1996), no. 10,
3217–3228. MR 1402555
(97j:16051a), http://dx.doi.org/10.1080/00927879608825743
 [By02]
Nigel
P. Byott, Integral HopfGalois structures on degree 𝑝²
extensions of 𝑝adic fields, J. Algebra 248
(2002), no. 1, 334–365. MR 1879021
(2002j:11142), http://dx.doi.org/10.1006/jabr.2001.9053
 [By04]
Nigel
P. Byott, HopfGalois structures on field extensions with simple
Galois groups, Bull. London Math. Soc. 36 (2004),
no. 1, 23–29. MR 2011974
(2004i:16049), http://dx.doi.org/10.1112/S0024609303002595
 [CaC99]
Scott
Carnahan and Lindsay
Childs, Counting Hopf Galois structures on nonabelian Galois field
extensions, J. Algebra 218 (1999), no. 1,
81–92. MR
1704676 (2000e:12010), http://dx.doi.org/10.1006/jabr.1999.7861
 [Ch00]
Lindsay
N. Childs, Taming wild extensions: Hopf algebras and local Galois
module theory, Mathematical Surveys and Monographs, vol. 80,
American Mathematical Society, Providence, RI, 2000. MR 1767499
(2001e:11116)
 [Ch03]
Lindsay
N. Childs, On Hopf Galois structures and complete groups, New
York J. Math. 9 (2003), 99–115. MR 2016184
(2004k:16097)
 [Ch05]
Lindsay
N. Childs, Elementary abelian Hopf Galois structures and polynomial
formal groups, J. Algebra 283 (2005), no. 1,
292–316. MR 2102084
(2005g:16073), http://dx.doi.org/10.1016/j.jalgebra.2004.07.009
 [Ch07]
Lindsay
N. Childs, Some Hopf Galois structures arising
from elementary abelian 𝑝groups, Proc.
Amer. Math. Soc. 135 (2007), no. 11, 3453–3460. MR 2336557
(2008j:16107), http://dx.doi.org/10.1090/S0002993907088880
 [CCo07]
Lindsay
N. Childs and Jesse
Corradino, Cayley’s theorem and Hopf Galois structures for
semidirect products of cyclic groups, J. Algebra 308
(2007), no. 1, 236–251. MR 2290920
(2007j:20026), http://dx.doi.org/10.1016/j.jalgebra.2006.09.016
 [Go82]
Daniel
Gorenstein, Finite simple groups, University Series in
Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to
their classification. MR 698782
(84j:20002)
 [GP87]
Cornelius
Greither and Bodo
Pareigis, Hopf Galois theory for separable field extensions,
J. Algebra 106 (1987), no. 1, 239–258. MR 878476
(88i:12006), http://dx.doi.org/10.1016/00218693(87)900299
 [Ko07]
Timothy
Kohl, Groups of order 4𝑝, twisted wreath products and
HopfGalois theory, J. Algebra 314 (2007),
no. 1, 42–74. MR 2331752
(2008e:12001), http://dx.doi.org/10.1016/j.jalgebra.2007.04.001
 [By96]
 N. P. Byott, Uniqueness of Hopf Galois structure of separable field extensions, Comm. Algebra 24 (1996), 32173228. MR 1402555 (97j:16051a)
 [By02]
 N. P. Byott, Integral HopfGalois structures on degree extensions of adic fields,
J. Algebra 248 (2002), 334365. MR 1879021 (2002j:11142)
 [By04]
 N. P. Byott, HopfGalois structures on field extensions with simple Galois groups, Bulletin of the London Mathematical Society 36 (2004), 2329. MR 2011974 (2004i:16049)
 [CaC99]
 S. Carnahan, L. N. Childs, Counting Hopf Galois structures on nonabelian Galois field extensions, J. Algebra 218 (1999), 8192. MR 1704676 (2000e:12010)
 [Ch00]
 L. N. Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, Mathematical Surveys and Monographs 80, American Mathematical Society, 2000. MR 1767499 (2001e:11116)
 [Ch03]
 L. N. Childs, On Hopf Galois structures and complete groups, New York J. Math. 9 (2003), 99115. MR 2016184 (2004k:16097)
 [Ch05]
 L. N. Childs, Elementary abelian Hopf Galois structures and polynomial formal groups, J. Algebra 283 (2005), 292316. MR 2102084 (2005g:16073)
 [Ch07]
 L. N. Childs, Some Hopf Galois structures arising from elementary abelian groups, Proc. Amer. Math. Soc. 135 (2007), 34533460. MR 2336557 (2008j:16107)
 [CCo07]
 L. N. Childs, J. Corradino, Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups, J. Algebra 308 (2007), 236251. MR 2290920 (2007j:20026)
 [Go82]
 D. Gorenstein, Finite Simple Groups, An Introduction to Their Classification, Plenum, New York/London, 1982. MR 698782 (84j:20002)
 [GP87]
 C. Greither, B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), 239258. MR 878476 (88i:12006)
 [Ko07]
 T. Kohl, Groups of order , twisted wreath products and HopfGalois theory, J. Algebra 314 (2007), 4274. MR 2331752 (2008e:12001)
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Additional Information
Lindsay N. Childs
Affiliation:
Department of Mathematics and Statistics, University at Albany, Albany, New York 12222
Email:
childs@math.albany.edu
DOI:
http://dx.doi.org/10.1090/S000299392012114182
PII:
S 00029939(2012)114182
Received by editor(s):
November 18, 2009
Received by editor(s) in revised form:
November 14, 2010, July 21, 2011, and August 25, 2011
Published electronically:
September 21, 2012
Additional Notes:
The author thanks the mathematics department at Virginia Commonwealth University for its hospitality while this research was conducted and the referee for numerous helpful suggestions.
Communicated by:
Ted Chinburg
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
