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Valuations on tensor powers of a division algebra

We study the following question in this paper: If p is a prime, m a positive integer, and S = (sm,...,s1) an arbitrary sequence consisting of "Y " or "N," does there exist a division algebra of exponent pm over a valued field (F , v) such that the underlying division algebra of the tensor power D⊗pi has a valuation extending v if and only if sm−i = Y ? We show that if such an algebra exists, then its index must be bounded below by a power of p that depends on both m and S, and we then answer the question affirmatively by constructing such an algebra of minimal index.

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