and -admissibility of groups of odd order

Authors:
Burton Fein and Murray Schacher

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1639-1645

MSC:
Primary 12E15; Secondary 16K40

MathSciNet review:
1242083

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the rational function field over the rationals, , let be the Laurent series field over , and let be a group of odd order. We investigate the following question: does there exist a finite-dimensional division algebra *D* central over or which is a crossed product for ? If such a *D* exists, is said to be -admissible (respectively, -admissible). We prove that if is -admissible, then is also -admissible; we also exhibit a -admissible group which is not -admissible.

**[FS]**Burton Fein and Murray Schacher,*Crossed products over algebraic function fields*, J. Algebra**171**(1995), no. 2, 531–540. MR**1315911**, 10.1006/jabr.1995.1026**[FSS]**Burton Fein, David J. Saltman, and Murray Schacher,*Crossed products over rational function fields*, J. Algebra**156**(1993), no. 2, 454–493. MR**1216479**, 10.1006/jabr.1993.1084**[JW]**Bill Jacob and Adrian Wadsworth,*Division algebras over Henselian fields*, J. Algebra**128**(1990), no. 1, 126–179. MR**1031915**, 10.1016/0021-8693(90)90047-R**[N]**Jürgen Neukirch,*On solvable number fields*, Invent. Math.**53**(1979), no. 2, 135–164. MR**560411**, 10.1007/BF01390030**[P]**Richard S. Pierce,*Associative algebras*, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. Studies in the History of Modern Science, 9. MR**674652****[Sc]**Murray M. Schacher,*Subfields of division rings. I*, J. Algebra**9**(1968), 451–477. MR**0227224****[Se]**Jean-Pierre Serre,*Local fields*, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR**554237****[Se]**-,*Cohomologie Galoisienne*, Lecture Notes in Math., vol. 5, Springer-Verlag, Berlin, Heidelberg, and New York, 1964.**[So]**Jack Sonn,*Rational division algebras as solvable crossed products*, Israel J. Math.**37**(1980), no. 3, 246–250. MR**599459**, 10.1007/BF02760966**[So]**Jack Sonn,*𝑄-admissibility of solvable groups*, J. Algebra**84**(1983), no. 2, 411–419. MR**723399**, 10.1016/0021-8693(83)90085-6**[W]**Edwin Weiss,*Algebraic number theory*, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR**0159805**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
12E15,
16K40

Retrieve articles in all journals with MSC: 12E15, 16K40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1242083-3

Keywords:
Division algebra,
Brauer group,
admissible,
crossed product

Article copyright:
© Copyright 1995
American Mathematical Society