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

 

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
Published electronically: October 11, 2000
MathSciNet review: 1814073
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Abstract: We consider Galois embedding problems $G\twoheadrightarrow H\cong \operatorname{Gal}(X/Z)$ such that a Galois embedding problem $G\twoheadrightarrow \operatorname{Gal}(Y/Z)$ is solvable, where $Y/Z$ is a Galois subextension of $X/Z$. For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative $k$-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 $p$-groups to which the reduction theorem applies.


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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: http://dx.doi.org/10.1090/S0002-9947-00-02625-8
PII: S 0002-9947(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