Realizability and automatic realizability of Galois groups of order 32

• Autorzy: Helen Grundman , Tara Smith
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Inverse Galois theory; 2-groups; Automatic realizability

Kategorie tematyczne

• 12F12: Inverse Galois theory

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 2
• Strony: 244-260

Daty

wydano

2010-04-01

online

2010-04-14

Twórcy

autor

Helen Grundman

• Bryn Mawr College

autor

Tara Smith

• University of Cincinnati

Bibliografia

• [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
• [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
• [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
• [6] Grundman H.G., Smith T.L., Swallow J. R., Groups of order 16 as Galois groups, Expo. Math., 1995, 13, 289–319
• [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
• [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
• [12] Ledet A., On 2-groups as Galois groups, Canad. J. Math., 1995, 47, 1253–1273
• [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
• [14] Michailov I., Embedding obstructions for the cyclic and modular 2-groups, Math. Balkanica (N.S.), 2007, 21, 31–50
• [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
• [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
• [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)

Identyfikatory

DOI

10.2478/s11533-009-0072-x