Is $Nm_{L/K}(L\backslash K)$ Zarisky dense in $\mathbb{A}^{1}(K)$? Here $Nm$ is the norm map for a finite field extension $L/K$. Note that we are excluding $K$ from $L$ before the map.
This is clearly true for finite field (in fact it is surjective). But for infinite field surjectivity might not be true, but is it still at least dense?