Given finite field extensions $K\subset E \subset F$, let $p\in E[x]$. Given any $\theta\in Gal(F/E)$, I know that $p(\theta(u))=0$ for any root $u$ of $p$ in $E$, i.e., $\theta$ permutes the roots of $p$ in $F$. My questions is:
Given $\theta\in Gal(F/K)$, does $\theta$ again permute the roots of $p$ in $F$?
I'm sorry if this is a stupid question. I am just learning Galois Theory, and this isn't homework, it is just something I can't sort out. Thank you for your help.
EDIT: What can we say if we know that the $\theta(a)\in F\setminus E$ for all $a\in F$?