Realizability and automatic realizability of Galois groups of order 32

Helen Grundman; Tara Smith

Open Mathematics (2010)

  • Volume: 8, Issue: 2, page 244-260
  • ISSN: 2391-5455

Abstract

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This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.

How to cite

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Helen Grundman, and Tara Smith. "Realizability and automatic realizability of Galois groups of order 32." Open Mathematics 8.2 (2010): 244-260. <http://eudml.org/doc/269614>.

@article{HelenGrundman2010,
abstract = {This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.},
author = {Helen Grundman, Tara Smith},
journal = {Open Mathematics},
keywords = {Inverse Galois theory; 2-groups; Automatic realizability; inverse Galois theory; automatic realizability; Brauer group; quaternion algebra},
language = {eng},
number = {2},
pages = {244-260},
title = {Realizability and automatic realizability of Galois groups of order 32},
url = {http://eudml.org/doc/269614},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Helen Grundman
AU - Tara Smith
TI - Realizability and automatic realizability of Galois groups of order 32
JO - Open Mathematics
PY - 2010
VL - 8
IS - 2
SP - 244
EP - 260
AB - This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.
LA - eng
KW - Inverse Galois theory; 2-groups; Automatic realizability; inverse Galois theory; automatic realizability; Brauer group; quaternion algebra
UR - http://eudml.org/doc/269614
ER -

References

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  1. [1] Carlson J., The mod 2 cohomology of 2-groups, Tables of 2-groups: 〈http://www.math.uga.edu/~lvalero/cohointro. html〉 
  2. [2] Gao W., Leep D. B., Minác J., Smith T. L., Galois groups over nonrigid fields, In: Valuation Theory and its Applications, Vol. II, Fields Institute Communications Series, 33, American Mathematical Society, 2003, 61–77 Zbl1049.12004
  3. [3] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.9, 2006, 〈http://www.gap-system.org〉 
  4. [4] Grundman H.G., Smith T.L., Automatic realizability of Galois groups of order 16, Proc. AMS, 1996, 124, 2631–2640 http://dx.doi.org/10.1090/S0002-9939-96-03345-X Zbl0862.12005
  5. [5] Grundman H.G., Smith T.L., Galois realizability of acentral C 4-extension of D 8, J. Alg., 2009, 322, 3492–3498 http://dx.doi.org/10.1016/j.jalgebra.2009.08.015 Zbl1222.12007
  6. [6] Grundman H.G., Smith T.L., Swallow J. R., Groups of order 16 as Galois groups, Expo. Math., 1995, 13, 289–319 Zbl0838.12004
  7. [7] Grundman H.G., Stewart G., Galois realizability of non-split group extensions of C 2 by (C 2)r × (C 4)s × (D 4)t, J. Algebra, 2004, 272, 425–434 http://dx.doi.org/10.1016/j.jalgebra.2003.09.017 Zbl1043.12004
  8. [8] Hall M.Jr., Senior J.K., The Groups of Order 2n (n ≤ 6), Macmillian, New York, 1964 
  9. [9] Ishkhanov V.V., Lur’e B.B., Faddeev D.K., The Embedding Problem in Galois Theory, Translations of Mathematical Monographs, 165, American Mathematical Society, Providence, R.I., 1997 
  10. [10] Jensen C.U., On the representation of a group as a Galois group over an arbitrary field, In: De Koninck J.-M., Levesque C. (Eds.), Theoriedes Nombres - Number Theory, Walterde Gruyter, 1989, 441–458 
  11. [11] Kuyk W., Lenstra H. W., Abelian extensions of arbitrary fields, Math. Ann., 1975, 216, 99–104 http://dx.doi.org/10.1007/BF01432536 Zbl0293.12102
  12. [12] Ledet A., On 2-groups as Galois groups, Canad. J. Math., 1995, 47, 1253–1273 Zbl0849.12006
  13. [13] Ledet A., Embedding problems with cyclic kernel of order 4, Israel J. Math., 1998, 106, 109–131 http://dx.doi.org/10.1007/BF02773463 Zbl0913.12004
  14. [14] Michailov I., Embedding obstructions for the cyclic and modular 2-groups, Math. Balkanica (N.S.), 2007, 21, 31–50 Zbl1219.12006
  15. [15] Michailov I., Groups of order 32 as Galois groups, Serdica Math. J., 2007, 33, 1–34 
  16. [16] Smith T.L., Extra-special 2-groups of order 32 as Galois groups, Canad. J. Math., 1994, 46, 886–896 Zbl0810.12004
  17. [17] Swallow J., Thiem N., Quadratic corestriction, C 2-embedding problems, and explicit construction, Comm. Algebra, 2002, 30, 3227–3258 http://dx.doi.org/10.1081/AGB-120004485 Zbl1019.12001
  18. [18] Witt E., Konstruktion von Galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung p f, J. Reine Angew. Math., 1936, 174, 237–245 (in German) 

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