Let $L|K$ be a finite normal extension. I want to show that for $x\in L$
$f(X)=\prod\limits_{\sigma\in \text{Gal}(L|K)}(X-\sigma(x))$ is in $K[X]$. If $L|K$ is a finite Galois extension, then the problem is over. But can it be shown for a weaker condition? The main issue is for a finite Galois extension $L|K$, the fixed field $\text{Fix}_{\text{Gal}(L|K)}L=K$. But is it true for a finite normal extension?
Any help is appreciated.
Why don't we think of an example of a non-separable extension, and test out this hypothesis?
How about taking $K = \mathbb F_p (t^n)$ and $L = \mathbb F_p(t)$, for some $n \in \mathbb N$?
So $L$ is the splitting field for the polynomial $X^n - t^n \in K[X]$, which factorises as $(X - t)^n$ in $L[X]$. (Hence the extension $L : K$ is a normal, but not separable.)
The group of automorphisms of $L$ fixing $K$ contains only the trivial automorphism, sending $t \mapsto t$. So if we define the polynomial $f(X)$ as in your question, with $x = t$, then $f(X) = (X - t)$, and this is not a polynomial in $K[X]$.