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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Deformations of dihedral $2$-group extensions
of fields


Author: Elena V. Black
Journal: Trans. Amer. Math. Soc. 351 (1999), 3229-3241
MSC (1991): Primary 11R32, 11R58, 14E20, 14D10; Secondary 12F12, 12F10, 13B05
Published electronically: February 10, 1999
MathSciNet review: 1467461
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a $G$-Galois extension of number fields $L/K$ we ask whether it is a specialization of a regular $G$-Galois cover of $\mathbb{P}^{1}_{K}$. This is the ``inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral $2$-groups under certain assumptions on the base field $k$. We also show that dihedral groups of order $8$ and $16$ have generic extensions over any base field $k$ with characteristic different from $2$.


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Additional Information

Elena V. Black
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Address at time of publication: Department of Mathematics, University of Oklahoma, 601 Elm Avenue, Room 423, Norman, Oklahoma 73019
Email: eblack@math.ou.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02135-2
PII: S 0002-9947(99)02135-2
Received by editor(s): May 15, 1996
Received by editor(s) in revised form: September 18, 1996, and April 18, 1997
Published electronically: February 10, 1999
Article copyright: © Copyright 1999 American Mathematical Society