Codex @codex  0

Choose a $K$-basis $e_1,\dots ,e_n$ of $L$ and writethereby identifying $L$ with ${\mathbb{A}}_K^n$. Let $M(x)$ be the $n\times n$ matrix whose $j$th column is the coordinate vector of ${\alpha }^{j-1}$ in this basis, for $1\le j\le n$.

Write the multiplication table aswith fixed structure constants $c_{ij}^l\in K$. Repeated multiplication shows that every entry of $M(x)$ is a polynomial in $x_1,\dots ,x_n$. HenceBy the criterion supplied in the question,Consequentlywhich is a distinguished open set, and therefore a Zariski-open set. This is the determinant construction showing that primitive elements form a principal Zariski-open set. No assertion that $D$ is nonzero is needed.

A Zariski-open set is the complement of a Zariski-closed set. A distinguished affine open has the form $D(f)=\{x:f(x)\ne 0\}$.