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Linkage of symbol p-algebras of degree 3

  • Adam ChapmanEmail author
Article
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Abstract

Given a field F of characteristic 3 and division symbol p-algebras \([\alpha ,\beta )_{3,F}\) and \([\alpha ,\gamma )_{3,F}\) of degree 3 over F, we prove that if \(\alpha {{\text {dlog}}}(\beta )\wedge {{\text {dlog}}}(\gamma )\) is trivial in the Kato–Milne cohomology group \(H_3^3(F)\), then the algebras share a common splitting field which is an inseparable degree 3 extension of either F or a quadratic extension of F. In the special case of quadratically closed fields, if \(\alpha {{\text {dlog}}}(\beta )\wedge {{\text {dlog}}}(\gamma )=0\), then they share an inseparable degree 3 extension of F.

Keywords

Kato–Milne cohomology Fields of positive characteristic Central simple algebras Division algebras Symbol algebras p-algebras Linkage 

Mathematics Subject Classification

Primary 16K20 Secondary 11E04 11E81 19D45 

Notes

Acknowledgements

The author thanks the anonymous referee for the careful reading of the submitted manuscript and the useful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceTel-Hai Academic CollegeUpper GalileeIsrael

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