I'm looking for division algebras over $\Bbb Q_{47}$. I guess my best bet is to calculate the Brauer group $\operatorname{Br}(\Bbb Q_{47})$. What's the best way of performing this calculation? Should I calculate $``$step by step$"$ $\operatorname{Br}(K/\Bbb Q_{47})$, where $K$ is a finite extension of $\Bbb Q_{47}$?
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4http://math.stackexchange.com/questions/321453/what-are-the-p-adic-division-algebras – Qiaochu Yuan May 05 '14 at 17:12