Abelian Hopf Galois structures on prime-power Galois field extensions

Authors:
S. C. Featherstonhaugh, A. Caranti and L. N. Childs

Journal:
Trans. Amer. Math. Soc. **364** (2012), 3675-3684

MSC (2010):
Primary 12F10; Secondary 16N20, 20B25, 16W30

Published electronically:
March 8, 2012

MathSciNet review:
2901229

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Abstract | References | Similar Articles | Additional Information

Abstract: The main theorem of this paper is that if is a finite abelian -group of -rank where , then every regular abelian subgroup of the holomorph of is isomorphic to . The proof utilizes a connection, observed by Caranti, Dalla Volta, and Sala, between regular abelian subgroups of the holomorph of and nilpotent ring structures on . Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let be a Galois extension of fields with abelian Galois group . If also is -Hopf Galois, where the -Hopf algebra has associated group with as above, then is isomorphic to .

**[By96]**N. P. Byott,*Uniqueness of Hopf Galois structure for separable field extensions*, Comm. Algebra**24**(1996), no. 10, 3217–3228. MR**1402555**, 10.1080/00927879608825743**[By04]**Nigel P. Byott,*Hopf-Galois structures on field extensions with simple Galois groups*, Bull. London Math. Soc.**36**(2004), no. 1, 23–29. MR**2011974**, 10.1112/S0024609303002595**[CDVS06]**A. Caranti, F. Dalla Volta, and M. Sala,*Abelian regular subgroups of the affine group and radical rings*, Publ. Math. Debrecen**69**(2006), no. 3, 297–308. MR**2273982****[CaC99]**Scott Carnahan and Lindsay Childs,*Counting Hopf Galois structures on non-abelian Galois field extensions*, J. Algebra**218**(1999), no. 1, 81–92. MR**1704676**, 10.1006/jabr.1999.7861**[CS69]**Stephen U. Chase and Moss E. Sweedler,*Hopf algebras and Galois theory*, Lecture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969. MR**0260724****[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****[Ch03]**Lindsay N. Childs,*On Hopf Galois structures and complete groups*, New York J. Math.**9**(2003), 99–115. MR**2016184****[Ch05]**Lindsay N. Childs,*Elementary abelian Hopf Galois structures and polynomial formal groups*, J. Algebra**283**(2005), no. 1, 292–316. MR**2102084**, 10.1016/j.jalgebra.2004.07.009**[Ch07]**Lindsay N. Childs,*Some Hopf Galois structures arising from elementary abelian 𝑝-groups*, Proc. Amer. Math. Soc.**135**(2007), no. 11, 3453–3460. MR**2336557**, 10.1090/S0002-9939-07-08888-0**[Fe03]**S. C. Featherstonhaugh, Abelian Hopf Galois extensions on Galois field extensions of prime power order, Ph.D. thesis, Univ. at Albany, NY, 2003.**[GP87]**Cornelius Greither and Bodo Pareigis,*Hopf Galois theory for separable field extensions*, J. Algebra**106**(1987), no. 1, 239–258. MR**878476**, 10.1016/0021-8693(87)90029-9**[HR07]**Christopher J. Hillar and Darren L. Rhea,*Automorphisms of finite abelian groups*, Amer. Math. Monthly**114**(2007), no. 10, 917–923. MR**2363058****[Ja65]**N. Jacobson,*Representation theory for Jordan rings*, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 37–43. MR**0044505****[Ko98]**Timothy Kohl,*Classification of the Hopf Galois structures on prime power radical extensions*, J. Algebra**207**(1998), no. 2, 525–546. MR**1644203**, 10.1006/jabr.1998.7479**[Ko07]**Timothy Kohl,*Groups of order 4𝑝, twisted wreath products and Hopf-Galois theory*, J. Algebra**314**(2007), no. 1, 42–74. MR**2331752**, 10.1016/j.jalgebra.2007.04.001

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Additional Information

**S. C. Featherstonhaugh**

Affiliation:
Department of Mathematics, Borough of Manhattan Community College/CUNY, 199 Chambers Street, Room N-520, New York, New York 10007

Email:
sfeatherstonhaugh@bmcc.cuny.edu

**A. Caranti**

Affiliation:
Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38123 Trento, Italy

Email:
caranti@science.unitn.it

**L. N. Childs**

Affiliation:
Department of Mathematics, University at Albany, Albany, New York 12222

Email:
lchilds@albany.edu

DOI:
https://doi.org/10.1090/S0002-9947-2012-05503-6

Received by editor(s):
July 1, 2010

Received by editor(s) in revised form:
August 24, 2010, and November 12, 2010

Published electronically:
March 8, 2012

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.