Automatic realizability of Galois groups of order 16
HTML articles powered by AMS MathViewer
- by Helen G. Grundman and Tara L. Smith
- Proc. Amer. Math. Soc. 124 (1996), 2631-2640
- DOI: https://doi.org/10.1090/S0002-9939-96-03345-X
- PDF | Request permission
Abstract:
This article examines the realizability of small groups of order $2^{k}, k \leq 4$, as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.References
- H. G. Grundman, T. L. Smith and J. Swallow, Groups of order 16 as Galois groups, Expo. Math. 13 (1995), 289–319.
- C. U. Jensen, On the representations of a group as a Galois group over an arbitrary field, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 441–458. MR 1024582
- C. U. Jensen, Finite groups as Galois groups over arbitrary fields, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 435–448. MR 1175848, DOI 10.24033/msmf.303
- W. Kuyk and H. W. Lenstra Jr., Abelian extensions of arbitrary fields, Math. Ann. 216 (1975), no. 2, 99–104. MR 424772, DOI 10.1007/BF01432536
- A. Ledet, On 2-groups as Galois groups, Canad. J. Math. 47 (1995), 1253–1273.
- Ján Mináč and Tara L. Smith, A characterization of $C$-fields via Galois groups, J. Algebra 137 (1991), no. 1, 1–11. MR 1090208, DOI 10.1016/0021-8693(91)90078-M
- Roger Ware, A note on the quaternion group as Galois group, Proc. Amer. Math. Soc. 108 (1990), no. 3, 621–625. MR 998741, DOI 10.1090/S0002-9939-1990-0998741-4
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- E. Witt, Konstruktion von galoisschen Körpern der Charakteristik $p$ zu vorgegebener Gruppe der Ordnung $p^{f}$, J. Reine Angew. Math. 174 (1936), 237–245.
Bibliographic Information
- Helen G. Grundman
- Affiliation: Department of Mathematics, Bryn Mawr College, Bryn Mawr, Pennsylvania 19010 and Mathematical Sciences Research Institute, Berkeley, California 94720
- MR Author ID: 307385
- Email: grundman@brynmawr.edu
- Tara L. Smith
- Affiliation: Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025
- Email: tsmith@math.uc.edu
- Received by editor(s): September 20, 1994
- Received by editor(s) in revised form: March 6, 1995
- Additional Notes: The first author’s research was supported in part by National Science Foundation Grant No. DMS-9115349 and the Alice Lee Hardenbergh Clark Faculty Research Grants Fund of Bryn Mawr College. The second author’s research was supported in part by the National Security Agency and the Taft Memorial Fund of the University of Cincinnati
- Communicated by: Lance W. Small
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2631-2640
- MSC (1991): Primary 12F10, 12F12
- DOI: https://doi.org/10.1090/S0002-9939-96-03345-X
- MathSciNet review: 1327017