Background and lemma first: Let $\Theta \in k[x_1,...,x_n]^H$ and $\theta$ be its evaluation to the roots of a fixed $n$:th degree polynomial in $k[x]$. Put $L(t) = \prod_{\sigma \in S_n//H} (t-\sigma \theta)$ ($[S_n : H]=e$), and let $E$ be the splitting field of said $n$:th degree polynomial. Lemma: If $\theta$ is a simple root of $L$ then $Gal(E/k(\theta))=G \cap H$, where $G=Gal(E/k)$.
Statement: Let $P$ (separable) be a simple irreducible factor over $k$ of $L$ of degree $j$. Then there is a subset $J$ of cardinality $j$ in $[e]$ such that $P = \prod_{m \in J} (t-\sigma_m \theta)$. For all $m \in J$ the degree $j$ of $P$ equals $[G : G \cap H]$ since $\theta_m$ is a simple root of $L$.
I don't understand at all how $[G:G\cap H]$ comes into the picture, or for that matter why the degree of $P$ is $[G:G\cap H]$?
edit: I realize the above is not a very good question. I have been going over this problem quite a bit, despite that my question remains of the type "what has that got to do with anything". Hopefully I can get an answer anyway.
Progress: Not much. I've looked over the basic results of GT but those results force me to introduce the fields $Ek(\theta_m)$, $F=spl_k(P)$ and possibly $EF$, but I'm unable to obtain any relevant information from this. So, I finally relented to trying a simpler situation, namely when the splitting field of $L$ is $E$ (this is true if $k$ is infinite and $H \neq S_n,A_n$).
In this case we simply get $k \subset k(\theta_m) \subset E$. By the tower law we then have
$[E:k] = [E:k(\theta_m)][k(\theta_m):k] \Rightarrow [k(\theta_m):k]=[E:k]/[E:k(\theta_m)] =^* |G|/|G \cap H_m|=[G:G \cap H_m]$
where $H_m = \sigma_m H \sigma_m^{-1}$. The degrees equal the orders of the resp. groups since both $E/k$ and $E/k(\theta_m)$ are Galois. I'm not entirely sure about the equality marked with an asterisk; the Lemma is stated as is, but $G \cap H = stab_G(H)$ ($H$ as coset of $S_n/H$) and $G \cap H_m = stab_G(\sigma_m H)$, so it seems reasonable. I'll check it rigorously "later".
I'd appreciate any input on this particular case (if something is wrong), and especially if anyone could give advice on how to show the general case, in which nothing is assumed about $k$ and hence the splitting field of $L$ is not necessarily $E$.
Progress 2: None, and I'm giving up on this one. Either I'm missing some elementary but critical fact, or this result is not trivial at all. The tower law, as it's stated everywhere I've seen it, holds for fields $k \subset F \subset E$, not for $k \subset E,F$ with the latter two contained in a larger field. Thus, the general result cannot follow from that. I've tried making explicit computations but have made no progress with that, nor can I find any helpful results.