Primitive elements form a principal Zariski-open set

Fix a $K$-basis of a finite extension $L/K$ of degree $n$. The coordinates of $1,\alpha ,\dots ,{\alpha }^{n-1}$ are polynomial functions of the coordinates of $\alpha$, because multiplication in $L$ has fixed structure constants. The determinant $D(\alpha )$ of those coordinate columns is therefore a polynomial, andThus the primitive-element locus is the distinguished Zariski-open set $D(D)\subseteq {\mathbb{A}}_K^n$, possibly empty for a non-simple extension.