## Deformations of dihedral 2-group extensions of fields

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- by Elena V. Black
- Trans. Amer. Math. Soc.
**351**(1999), 3229-3241 - DOI: https://doi.org/10.1090/S0002-9947-99-02135-2
- Published electronically: February 10, 1999
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## 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 â€śinverse" 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 $2$.## References

- Allen Altman and Steven Kleiman,
*Introduction to Grothendieck duality theory*, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR**0274461**, DOI 10.1007/BFb0060932 - Sybilla Beckmann,
*Is every extension of $\textbf {Q}$ the specialization of a branched covering?*, J. Algebra**164**(1994), no.Â 2, 430â€“451. MR**1271246**, DOI 10.1006/jabr.1994.1068 - E. Black,
*Arithmetic lifting of dihedral extensions*, J. of Algebra**203**(1998), 12â€“29. - A. Grothendieck,
*SĂ©minaire de la GĂ©omĂ©trie AlgĂ©brique: RevĂŞtements Etales et Groupe Fondamental*, Springer-Verlag, Heidelberg, 1971. - Nathan Jacobson,
*Basic algebra. II*, W. H. Freeman and Co., San Francisco, Calif., 1980. MR**571884** - G. Kemper,
*Generic Polynomials and Noetherâ€™s Problem for Linear Groups*, IWR preprint 95-19, Heidelberg (1995). - Ian Kiming,
*Explicit classifications of some $2$-extensions of a field of characteristic different from $2$*, Canad. J. Math.**42**(1990), no.Â 5, 825â€“855. MR**1080998**, DOI 10.4153/CJM-1990-043-6 - David J. Saltman,
*Generic Galois extensions and problems in field theory*, Adv. in Math.**43**(1982), no.Â 3, 250â€“283. MR**648801**, DOI 10.1016/0001-8708(82)90036-6 - David J. Saltman,
*Groups acting on fields: Noetherâ€™s problem*, Group actions on rings (Brunswick, Maine, 1984) Contemp. Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp.Â 267â€“277. MR**810657**, DOI 10.1090/conm/043/810657 - Richard G. Swan,
*Invariant rational functions and a problem of Steenrod*, Invent. Math.**7**(1969), 148â€“158. MR**244215**, DOI 10.1007/BF01389798

## Bibliographic 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
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 3229-3241 - MSC (1991): Primary 11R32, 11R58, 14E20, 14D10; Secondary 12F12, 12F10, 13B05
- DOI: https://doi.org/10.1090/S0002-9947-99-02135-2
- MathSciNet review: 1467461