论文标题
符号代数的常见分裂字段
Common Splitting Fields of Symbol Algebras
论文作者
论文摘要
我们研究了$ p^m $ $ p^m $ $ f $ $ f $ $ \ operatatorName {char}（f）= p $的符号代数的常见分裂字段。我们首先表明，如果任何有限数量的此类代数共享一个学位$ p^m $简单纯粹是不可分割的分裂字段，则它们共享一个相同程度的循环分裂字段。结果，我们得出的结论是，每个有限数量的符号代数$ p^{m_0}，\ dots，p^{m_t} $共享一个循环分裂$ p^{m_0+\ dots+dots+m_t} $。该概括恢复了以下事实，即符号代数的每个张量产物都是符号代数。我们应用Tignol的结果来绑定$ \ operatorname {br} _ {p^m}（f）$中的类符号长度，其符号长度嵌入到$ \ operatorname {br} _ {p^{p^{m+1}}}（f）（f）$ 2中是$ p \ in \ in \ in \ in \ {2,3，3，3.3，3，$ p \}。我们还研究了其他Kato-Milne共同体学组中的类似情况，在该小组中存在必要的规范条件。
We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. We first show that if any finite number of such algebras share a degree $p^m$ simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees $p^{m_0},\dots,p^{m_t}$ share a cyclic splitting field of degree $p^{m_0+\dots+m_t}$. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol's to bound the symbol length of classes in $\operatorname{Br}_{p^m}(F)$ whose symbol length when embedded into $\operatorname{Br}_{p^{m+1}}(F)$ is 2 for $p\in \{2,3\}$. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.
