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.
@article{bwmeta1.element.doi-10_2478_s11533-009-0072-x,
author = {Helen Grundman and Tara Smith},
title = {Realizability and automatic realizability of Galois groups of order 32},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {244-260},
zbl = {1256.12002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0072-x}
}
Helen Grundman; Tara Smith. Realizability and automatic realizability of Galois groups of order 32. Open Mathematics, Tome 8 (2010) pp. 244-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0072-x/
[1] Carlson J., The mod 2 cohomology of 2-groups, Tables of 2-groups: 〈http://www.math.uga.edu/~lvalero/cohointro. html〉
[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 | Zbl 1049.12004
[3] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.9, 2006, 〈http://www.gap-system.org〉
[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 | Zbl 0862.12005
[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 | Zbl 1222.12007
[6] Grundman H.G., Smith T.L., Swallow J. R., Groups of order 16 as Galois groups, Expo. Math., 1995, 13, 289–319 | Zbl 0838.12004
[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 | Zbl 1043.12004
[8] Hall M.Jr., Senior J.K., The Groups of Order 2n (n ≤ 6), Macmillian, New York, 1964
[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] 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] Kuyk W., Lenstra H. W., Abelian extensions of arbitrary fields, Math. Ann., 1975, 216, 99–104 http://dx.doi.org/10.1007/BF01432536 | Zbl 0293.12102
[12] Ledet A., On 2-groups as Galois groups, Canad. J. Math., 1995, 47, 1253–1273 | Zbl 0849.12006
[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 | Zbl 0913.12004
[14] Michailov I., Embedding obstructions for the cyclic and modular 2-groups, Math. Balkanica (N.S.), 2007, 21, 31–50 | Zbl 1219.12006
[15] Michailov I., Groups of order 32 as Galois groups, Serdica Math. J., 2007, 33, 1–34
[16] Smith T.L., Extra-special 2-groups of order 32 as Galois groups, Canad. J. Math., 1994, 46, 886–896 | Zbl 0810.12004
[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 | Zbl 1019.12001
[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)