Let $K$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$. Let us consider the polynomial ring $R=K[x_1,x_2,...,x_n]$ in $n$-variables and $f_1, f_2, \cdots, f_m \in K[x_1, \cdots, x_n]$.
Consider a finite extension $L$ of $K$ and consider the following zero set: $$S=\{f_i(x_1, \cdots, x_n)=0, \ i=1,\cdots,m \},$$ where each $f_i \in L[x_1,x_2, \cdots , x_n]$. Then, clearly every element in $S$ is a point in an affine $n$-space over $L$. So the coordinates of each points in $S$ generate a field extension. i.e., conisder the field extension $L(S)$ obtained adjoining the coordinates of each solutions in $S$.
Questions:
$(1)$ Is (or When) the extension $L(S)/K$ Galois ?
$(2)$ Is (or When) the extension $L(S)/L$ Galois ?
$(3)$ When are the above two extensions totally ramified ?
$$-----------$$ $(1)$ If we assume $L$ is an unramified extension of $K$, then $L/K$ is Galois extension. Now $L(S)$ is the algebraic extension of $L$ because its elements are algebraic over $L$. Thus $L(S)/L$ is also Galois extension. Hence $L(S)/K$ is Galois extension.
Am I correct ?
What about the other two questions ?
Any discussion please