Need help proving L is a normal exension of F.

[ae1709]

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

 

[micromass]

So, what did you try already?? (1) shouldn't be too difficult. It's just checking that something is a field...