Abstract
We study the common splitting fields of symbol algebras of degree \(p^m\) over fields F of \({\text {char}}(F)=p\). 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 \({\text {Br}}_{p^m}(F)\) whose symbol length when embedded into \({\text {Br}}_{p^{m+1}}(F)\) 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.
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Acknowledgements
The authors thank Eliyahu Matzri for bringing to our attention important works in the literature, which improved the quality of the paper. The first author acknowledges the receipt of the Chateaubriand Fellowship (969845L) offered by the French Embassy in Israel in the fall of 2020, which helped establish the scientific connection with the second author. The third author was supported by a grant from the Simons Foundation (580782).
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Chapman, A., Florence, M. & McKinnie, K. Common splitting fields of symbol algebras. manuscripta math. 171, 649–662 (2023). https://doi.org/10.1007/s00229-022-01401-2
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DOI: https://doi.org/10.1007/s00229-022-01401-2