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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1242083-3
PII: S 0002-9939(1995)1242083-3
Keywords: Division algebra, Brauer group, admissible, crossed product
Article copyright: © Copyright 1995 American Mathematical Society