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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 ?

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    Why is $L/K$ Galois?2012-12-09
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    It is trivial that $g$ splits in $L$, but I guess you also must show it doesn't split over any smaller field.2012-12-09
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    1. No. Separability may fail, for example if $K$ has char $p$, $u_1, u_2$ are purely inseparable over $K$, then you still have the inseparability issue for $L/K(v_0,\cdots,v_k)$.2012-12-09
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    2. yes. An automorphism of $L$ fixing $K$ will permute the $u_i$, and since $v_0,\cdots,v_k$ are symmetric polynomials in terms of $u_i$, they would be fixed under any such permutation.2012-12-09
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    @Sanchez I don't understand, the explanations are too terse (and I'm too much of a beginner to get much from them). Could you please expand your comments to an answer ? (And could you provide a concrete example for 1. please ?)2012-12-09
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    @user47574, do you know about inseparable extensions? What is your definition of Galois extensions?2012-12-09

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