Let $L/K$ be a field extension such that $L$ is a splitting field of $f\in K[X]$, i.e. $f=\prod_{i=1}^{k} (X-u_i)^{n_i}$ for some $u_i\in L$. If we denote the coefficients of $g:=\prod_{i=1}^{k} (X-u_i)$ with $v_0,\ldots,v_k$ is then
1) $L/K(v_0,\ldots,v_k)$ a Galois extension ?
2) Does $\text{Gal}(L/K(v_0,\ldots,v_k))=\text{Gal}(L/K) $ hold ?
I also have to show that $L$ is a splitting field of $g$, but this seems trivial, since $L$ contains the $u_i$'s, so if $f$ split, also $g$ has to split - or am I missing something here ?