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I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...

Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...