= Primitive elements form a principal Zariski-open set
Fix a $K$-basis of a finite extension $L/K$ of degree $n$. The coordinates of $1,\alpha,\ldots,\alpha^{n-1}$ 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, and
$$
K[\alpha]=L\quad\Longleftrightarrow\quad D(\alpha)\ne0.
$$
Thus the primitive-element locus is the distinguished <Zariski-open set> $D(D)\subseteq\mathbb A_K^n$, possibly empty for a non-simple extension.
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