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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$\textbf {Q}(t)$ and $\textbf {Q}((t))$-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
DOI: https://doi.org/10.1090/S0002-9939-1995-1242083-3
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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Keywords: Division algebra, Brauer group, admissible, crossed product
Article copyright: © Copyright 1995 American Mathematical Society