I want to understand the proof presented in the book. I figured out a way not to use the proof mentioned in the book though it is equivalent to what book says.($E/K$ is normal iff any automorphism of $C/K$ restricts to an autmorphism of $E/K$ iff $E/K$ is a splitting field of $M\subset K[x]$.)
Suppose $E/K$ is normal and $K'/K$ any field extension. Then $EK'/K'$ is normal.(This is more or less translation theorem if one assumes $E/K$ galois.)
The book says the following.
Since $E/K$ is normal, $E$ the splitting field of $M\subset K[x]$. Then $EK'$ is the splitting field of $M\subset K'[x]$. Let $L$ be the splitting field of $M\subset K'[x]$.
$\textbf{Q:}$ I used field automorphism to bypass the whole pf above to see $EK'/K'$ is normal. It is clear that $L\subset EK'$. How do I see $EK'\subset L$?(One should not use field automorphism here.)
The roots of a polynomial $P\in M$ are in $E\subset K'E$ so $M$ splits in $K'E$. If $a_1,...a_n,...$ are the elements of $M$, $K'E=K'(a_1,...,a_n)$.