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## Main Question or Discussion Point

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