Let $K$ be a complete discretely valued field of rank one, with residue field $\Q_p$. It is well known that period equals index in $\Br(K)$. We prove that when $p=2$ there exist noncyclic $K$-division algebras of every $2$-power degree divisible by four. Otherwise, every $K$-division algebra is cyclic.

中文翻译：

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布劳尔一维域上的非循环除法代数

令 $K$ 是一个完整的秩为离散值的字段，带有残差字段 $\Q_p$。众所周知，周期等于 $\Br(K)$ 中的索引。我们证明当 $p=2$ 时，每个 $2$-幂次都存在可被 4 整除的非循环 $K$-除法代数。否则，每个 $K$-division 代数都是循环的。