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

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

Published electronically:
March 8, 2012

MathSciNet review:
2901229

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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 .

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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.