We let $E/F$ be an algebraic extension of a field $F$, and define $S$ to be the set of all $F$-conjugates of the elements of $E$. Now we define $F(S)$ the be the field generated by $S$ over $F$. rove:
$1$.$F(S)/F$ is a normal extension.
$2$.$E\subseteq F(S)$.
Now for some reason I cant wrap my head around the definition of normal extensions. I tried solving the second item as follows:
$2$. Let $\alpha\in E$. I want to prove that $\alpha\in F(S)$. $\alpha\in E$ and therefore $\alpha$ is algebraic over $F$, meaning there exists a minimal irreduciable polynomial $f(x)$ over $F$ s.t $\alpha$ is its root. Since $\alpha$ is a $F$-conjugation of itself. by definition of $S$ its clear that $\alpha\in F(S)$.
Is my proof correct?
Furthermore, how do I approach the first item? I feel like it's not hard but I'm missing something.
Any help would be appreciated.