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Fields of characteristic $p$, i.e., fields for which there is a prime $p$ such that $px=0$ for each $x$. Do not use this tag for questions on characteristic polynomials of a matrix.
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Inseparable Galois Cohomology
I cannot answer your question, but point to the right algebraic framework in my opinion:
There is a well worked out classical (but somewhat underestimated) theory of inseparable Galois extensions. It …