Given $F\subset$ $L\subset$ $K$ being fields in $\Bbb{C}$, with $K:F$ and $L:F$ being normal extensions, with $\sigma$ $\in$ $G$, the galois group for $K:F$. Prove for $l\in$ $L$, $\sigma(l)\in$ $L$.
I'm guessing I have to use the fact that it's a normal extension, but I'm not sure how that ties to the properties of automorphisms. From what I understand, a normal extension means that for all irreducible polynomials in $F$, they either reduce into linear factors in $K$, or are still irreducible.
Let $P=a_0+..+a_nX^n$be the minimal polynomial of $l$ over $F$. $P^{\sigma}=\sigma(a_0)+..\sigma(a_n)X^n=P$ since $\sigma\in G=Gal(K|F)$ thus $\sigma(a_i)=a_i$. We can also write $P=(X-x_1)..(X-x_n) x_n\in L$ since $L$ is normal, and $P^{\sigma}=(X-\sigma(x_1))...(X-\sigma(x_n))=P$. This implies that $\sigma(x_i)=x_j\in L$. In particular $\sigma(l)=x_{i_0}$ since $l$ is a root of $P$.