$\textbf {Q}(t)$ and $\textbf {Q}((t))$-admissibility of groups of odd order

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.