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Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

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p-powers in a purely inseparable extension

Let $k \subset K \subset L$ be finite purely inseparable extensions of characteristic $p >0$. I am interested in knowing whether the group $$ \frac{(kK^p)^\ast \cap (k^\ast {L^p}^\ast)}{k^\ast {K^p}^\ …