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Let
\begin{equation}
{F}\subseteq{K_i}\subseteq{E}
\end{equation}
for i = 1,...,r with E a finite extension of F and where the intermediate fields \begin{equation} K_i \end{equation}are each normal extensions of F for all i.
Define:
\begin{equation}
L = \{f(a_1,....,a_r) : [f]\in{F[x]}, [a_i]\in[K_i]\}
\end{equation}
1) Prove that L is a subfield of E and contains \begin{equation} K_i \end{equation} for all i.
2) Prove that L is a normal extension of F.
I really need help with this. Thanks
\begin{equation}
{F}\subseteq{K_i}\subseteq{E}
\end{equation}
for i = 1,...,r with E a finite extension of F and where the intermediate fields \begin{equation} K_i \end{equation}are each normal extensions of F for all i.
Define:
\begin{equation}
L = \{f(a_1,....,a_r) : [f]\in{F[x]}, [a_i]\in[K_i]\}
\end{equation}
1) Prove that L is a subfield of E and contains \begin{equation} K_i \end{equation} for all i.
2) Prove that L is a normal extension of F.
I really need help with this. Thanks