Automatic realizability

of Galois groups of order 16

Authors:
Helen G. Grundman and Tara L. Smith

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

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Abstract | References | Similar Articles | Additional Information

Abstract: This article examines the realizability of small groups of order , as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.

**[GSS:1995]**H. G. Grundman, T. L. Smith and J. Swallow,*Groups of order 16 as Galois groups*, Expo. Math.**13**(1995), 289--319.**[Je:1989]**C. U. Jensen,*On the representations of a group as a Galois group over an arbitrary field*, Théorie des nombres -- Number Theory (J.-M. De Koninck and C. Levesque, eds.), Walter de Gruyter, 1989, pp. 441--458. MR**90k:12006****[Je:1992]**------,*Finite groups as Galois groups over arbitrary fields*, Cont. Math.**131**(1992), 534--448. MR**93i:12008****[KuLe:1975]**W. Kuyk and H. W. Lenstra, Jr.,*Abelian extensions of arbitrary fields*, Math. Ann.**216**(1975), 99--104. MR**54:12730****[L:1995]**A. Ledet,*On 2-groups as Galois groups*, Canad. J. Math.**47**(1995), 1253--1273.**[MiSm:1991]**J. Miná\v{c} and T. L. Smith,*A characterization of C-fields via Galois groups*, J. Algebra**137**(1991), 1--11. MR**92c:11033****[Wa:1990]**R. Ware,*A note on the quaternion group as Galois group*, Proc. Amer. Math. Soc.**108**(1990), 621--625. MR**90g:12006****[Wh:1957]**G. Whaples,*Algebraic extensions of arbitrary fields*, Duke Math. J.**24**(1957), 201--204. MR**19:8b****[Wi:1936]**E. Witt,*Konstruktion von galoisschen Körpern der Charakteristik zu vorgegebener Gruppe der Ordnung*, J. Reine Angew. Math.**174**(1936), 237--245.

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Additional Information

**Helen G. Grundman**

Affiliation:
Department of Mathematics, Bryn Mawr College, Bryn Mawr, Pennsylvania 19010 and Mathematical Sciences Research Institute, Berkeley, California 94720

Email:
grundman@brynmawr.edu

**Tara L. Smith**

Affiliation:
Department of Mathematics, University of Cincinnati, Cincinnati, Ohio 45221-0025

Email:
tsmith@math.uc.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03345-X

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

Article copyright:
© Copyright 1996
American Mathematical Society