I am reading Kaplansky's notes on Galois theory. He defines a normal extension as follows:
Let $M$ be any field and $K$ any subfield. $M$ is normal over $K$ if for any $u\in M$ but not in $K$, there exists an automorphism of $M$ leaving every element of $K$ fixed but actually moving $U$.
As a side note, this definition does not appear on Wikipedia. Could someone verify it is equivalent to the normal definitions?
He sets the following exercise:
Problem: Let $K\subset L \subset M$ be fields with $L$ normal over $K$ and $M$ normal over $L$. Suppose any automorphism of $L/K$ can be extended to $M$. Prove that $M$ is normal over $K$.
My solution attempt is in the answers. I also wonder how any automorphism of $L/K$ can be extended to $M$.