𝑄(𝑡) and 𝑄((𝑡))-admissibility of groups of odd order

Abstract

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