Let $L/K/F$ be a tower of separable finite field extensions and let $x\in L$ such that $L=K(x)$. Under which conditions is it possible to choose $x$ such that $N_{L/K}(x)$ is again a primitive generator of $K/F$? In other words, when is it possible to choose $x$ in general such that $L=K(x)$ and $K=F(N_{L/K}(x))$? Are there different conditions in case one takes the trace instead of the norm?
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