Relative Embedding Problems
Authors:
Elena V. Black and John R. Swallow
Journal:
Trans. Amer. Math. Soc. 353 (2001), 23472370
MSC (2000):
Primary 12F12, 13B05; Secondary 12F10
Published electronically:
October 11, 2000
MathSciNet review:
1814073
Fulltext PDF Free Access
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]
Elena
V. Black, Deformations of dihedral 2group
extensions of fields, Trans. Amer. Math.
Soc. 351 (1999), no. 8, 3229–3241. MR 1467461
(99m:12004), http://dx.doi.org/10.1090/S0002994799021352
 [CHR]
S.
U. Chase, D.
K. Harrison, and Alex
Rosenberg, Galois theory and Galois cohomology of commutative
rings, Mem. Amer. Math. Soc. No. 52 (1965),
15–33. MR
0195922 (33 #4118)
 [Cr]
Teresa
Crespo, Galois representations, embedding problems and modular
forms, Collect. Math. 48 (1997), no. 12,
63–83. Journées Arithmétiques (Barcelona, 1995). MR 1464017
(98j:11101)
 [DI]
Frank
DeMeyer and Edward
Ingraham, Separable algebras over commutative rings, Lecture
Notes in Mathematics, Vol. 181, SpringerVerlag, BerlinNew York, 1971. MR 0280479
(43 #6199)
 [GSS]
Helen
G. Grundman, Tara
L. Smith, and John
R. Swallow, Groups of order 16 as Galois groups, Exposition.
Math. 13 (1995), no. 4, 289–319. MR 1358210
(96h:12005)
 [Ho]
Klaus
Hoechsmann, Zum Einbettungsproblem, J. Reine Angew. Math.
229 (1968), 81–106 (German). MR 0244190
(39 #5507)
 [Ik]
Masatoshi
Ikeda, Zur Existenz eigentlicher galoisscher Körper beim
Einbettungsproblem für galoissche Algebren, Abh. Math. Sem. Univ.
Hamburg 24 (1960), 126–131 (German). MR 0121364
(22 #12103)
 [ILF]
V.
V. Ishkhanov, B.
B. Lur′e, and D.
K. Faddeev, The embedding problem in Galois theory,
Translations of Mathematical Monographs, vol. 165, American
Mathematical Society, Providence, RI, 1997. Translated from the 1990
Russian original by N. B. Lebedinskaya. MR 1454614
(98c:12007)
 [Le]
Arne
Ledet, Embedding problems with cyclic kernel of order 4,
Israel J. Math. 106 (1998), 109–131. MR 1656869
(99k:12009), http://dx.doi.org/10.1007/BF02773463
 [Li]
Steven
Liedahl, Presentations of metacyclic 𝑝groups with
applications to 𝐾admissibility questions, J. Algebra
169 (1994), no. 3, 965–983. MR 1302129
(96a:20043), http://dx.doi.org/10.1006/jabr.1994.1321
 [Sc]
Leila
Schneps, On cyclic field extensions of degree 8, Math. Scand.
71 (1992), no. 1, 24–30. MR 1216101
(94d:12004)
 [Sw1]
John
R. Swallow, Embedding problems and the
𝐶₁₆→𝐶₈ obstruction, Recent
developments in the inverse Galois problem (Seattle, WA, 1993), Contemp.
Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995,
pp. 75–90. MR 1352268
(96j:12010), http://dx.doi.org/10.1090/conm/186/02177
 [Sw2]
John
R. Swallow, Solutions to central embedding problems are
constructible, J. Algebra 184 (1996), no. 3,
1041–1051. MR 1407883
(97e:12007), http://dx.doi.org/10.1006/jabr.1996.0297
 [Bl]
 E. Black, Deformation of dihedral group extensions of fields, Trans. Amer. Math. Soc. 351 (1999), 32293241.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), 6383. MR 98j:11101
 [DI]
 F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, vol. 181, SpringerVerlag, Berlin, 1971. MR 43:6199
 [GSS]
 H. Grundman, T. Smith, and J. Swallow, Groups of order 16 as Galois groups, Exposition. Math. 13 (1995), 289319. MR 96h:12005
 [Ho]
 K. Hoechsmann, Zum Einbettungsproblem, J. Reine Angew. Math. 229 (1968), 81106. MR 39:5507
 [Ik]
 M. Ikeda, Zur Existenz eigentlicher galoisscher Körper beim Einbettungsproblem für galoissche Algebren, Abh. math. Sem. Hamburg 24 (1960), 126131. 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), 109132. MR 99k:12009
 [Li]
 S. Liedahl, Presentations of metacyclic groups with applications to admissibility questions, J. Algebra 169 (1994), 965983. MR 96a:20043
 [Sc]
 L. Schneps, On cyclic field extensions of degree 8, Math. Scand. 71 (1992), 2430. 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. 7590. MR 96j:12010
 [Sw2]
 J. Swallow, Solutions to central embedding problems are constructible, J. Algebra 184 (1996), 10411051. MR 97e:12007
Similar Articles
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:
http://dx.doi.org/10.1090/S0002994700026258
PII:
S 00029947(00)026258
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.\ DMS9501366 and a Davidson College MacArthur Faculty Study and Research Grant.
Article copyright:
© Copyright 2000
American Mathematical Society
