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I was looking for a possible reference that would answer the following question,

Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over $\mathbb{Q}_{p}$. Does anyone know of a reference that addresses the following question,

"Are all division algebras over $\mathbb{Q}_{p}((t))$ cyclic?"

Thanks in advance.

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  • $\begingroup$ By "division algebra" I assume you mean a finite dimensional central algebra over $\mathbb{Q}_p((t))$, which is a divsion ring. Is that correct? $\endgroup$ – Pace Nielsen Apr 7 '15 at 1:28
  • $\begingroup$ Yes, that is exactly what I mean by that. $\endgroup$ – TheNumber23 Apr 7 '15 at 1:41
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Here is the closest result I know. If the degree of the division algebra is a prime $q \ne p$, an affirmative answer has been given over finite extensions of $\mathbb Q_p(t)$ by Saltman's paper Cyclic algebras over $p$-adic curves.

There is also an article of Brussels on Saltman's work.

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  • $\begingroup$ Saltman's paper is working with the arithmetically trickier $\mathbb{Q}_{p}(t)$. But it is a good place to start. Thanks. $\endgroup$ – TheNumber23 Apr 7 '15 at 15:01
  • $\begingroup$ @TheNumber23 Ah, sorry, I had function fields on my mind when I saw your question, and didn't read it carefully. $\endgroup$ – Kimball Apr 7 '15 at 15:48

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