
TL;DR
This paper investigates conditions under which certain degree 3 symbol p-algebras over fields of characteristic 3 share common splitting fields, linking cohomological triviality to algebraic extensions.
Contribution
It establishes a connection between Kato-Milne cohomology triviality and the existence of inseparable degree 3 splitting fields for these algebras.
Findings
If the cohomology class is trivial, the algebras share an inseparable degree 3 splitting field.
Over quadratically closed fields, trivial cohomology implies a shared inseparable degree 3 extension.
The results relate cohomological conditions to explicit algebraic extensions in characteristic 3.
Abstract
Given a field of characteristic and division symbol -algebras and of degree over , we prove that if is trivial in the Kato-Milne cohomology group then the algebras share a common splitting field which is an inseparable degree 3 extension of either or a quadratic extension of . In the special case of quadratically closed fields, if , then they share an inseparable degree 3 extension of .
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