Given a finite separable extension $E \subset F \subset \bar E$, I am asked to construct
- A normal extension $M \subset \bar E$ such that $[M:E] \leq [F:E]!$
- a unique minimal normal extension $M \subset \bar E$ containing $E$.
$E \subset F$ finite means that it is finitely generated, then $F = E(a_1, \ldots, a_n)$ and $n$ can be taken to be $[F：E]$. Then let $m_i$ be the minimal polynomial of $a_i$. $\prod m_i$ is separable by the uniqueness of minimal polynomials. Then take $M$ to be the splitting field of $\prod m_i$, we get a normal extension over $F$. However how do I make it satisfy each case listed above?
EDIT: should be the minimal normal extension containing $F$.