Relative Embedding Problems

Authors:
Elena V. Black and John R. Swallow

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2347-2370

MSC (2000):
Primary 12F12, 13B05; Secondary 12F10

DOI:
https://doi.org/10.1090/S0002-9947-00-02625-8

Published electronically:
October 11, 2000

MathSciNet review:
1814073

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider Galois embedding problems such that a Galois embedding problem is solvable, where is a Galois subextension of . For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative -algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic -groups to which the reduction theorem applies.

**[Bl]**E. Black,*Deformation of dihedral**-group extensions of fields*, Trans. Amer. Math. Soc.**351**(1999), 3229-3241.MR**99m:12004****[CHR]**S. Chase, D. Harrison, and A. Rosenberg,*Galois theory and Galois cohomology of commutative rings*, Mem. Amer. Math. Soc., vol. 52, American Mathematical Society, Providence, RI, 1965; reprinted with corrections, 1968.MR**33:4118****[Cr]**T. Crespo,*Galois representations, embedding problems and modular forms*, Journées Arithmétiques (Barcelona, 1995), Collect. Math.**48**(1997), 63-83. MR**98j:11101****[DI]**F. DeMeyer and E. Ingraham,*Separable algebras over commutative rings*, Lecture Notes in Mathematics, vol. 181, Springer-Verlag, Berlin, 1971. MR**43:6199****[GSS]**H. Grundman, T. Smith, and J. Swallow,*Groups of order 16 as Galois groups*, Exposition. Math.**13**(1995), 289-319. MR**96h:12005****[Ho]**K. Hoechsmann,*Zum Einbettungsproblem*, J. Reine Angew. Math.**229**(1968), 81-106. MR**39:5507****[Ik]**M. Ikeda,*Zur Existenz eigentlicher galoisscher Körper beim Einbettungsproblem für galoissche Algebren*, Abh. math. Sem. Hamburg**24**(1960), 126-131. MR**22:12103****[ILF]**I. Ishkhanov, B. Lur'e, and D. Faddeev,*The embedding problem in Galois theory*, Translations of Mathematical Monographs, vol. 165, American Mathematical Society, Providence, RI, 1997. MR**98c:12007****[Le]**A. Ledet,*Embedding problems with cyclic kernel of order 4*, Israel J. Math.**106**(1998), 109-132. MR**99k:12009****[Li]**S. Liedahl,*Presentations of metacyclic**-groups with applications to**-admissibility questions*, J. Algebra**169**(1994), 965-983. MR**96a:20043****[Sc]**L. Schneps,*On cyclic field extensions of degree 8*, Math. Scand.**71**(1992), 24-30. MR**94d:12004****[Sw1]**J. Swallow,*Embedding problems and the**obstruction*, Contemporary Mathematics 186: Recent Developments in the Inverse Galois Problem, American Mathematical Society, Providence, RI, 1995, pp. 75-90. MR**96j:12010****[Sw2]**J. Swallow,*Solutions to central embedding problems are constructible*, J. Algebra**184**(1996), 1041-1051. MR**97e:12007**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
12F12,
13B05,
12F10

Retrieve articles in all journals with MSC (2000): 12F12, 13B05, 12F10

Additional Information

**Elena V. Black**

Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019

Address at time of publication:
131 Salina Street, Lafayette, Colorado 80026

Email:
eblack@math.ou.edu

**John R. Swallow**

Affiliation:
Department of Mathematics, Davidson College, Davidson, North Carolina 28036

Email:
joswallow@davidson.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02625-8

Received by editor(s):
January 4, 1999

Received by editor(s) in revised form:
August 20, 1999

Published electronically:
October 11, 2000

Additional Notes:
The first author gratefully acknowledges a University of Oklahoma Junior Faculty Research Grant. The second author gratefully acknowledges support under National Science Foundation Grant No. DMS-9501366 and a Davidson College MacArthur Faculty Study and Research Grant.

Article copyright:
© Copyright 2000
American Mathematical Society