Questions regarding tower of normal/separable extensions

Question

I am learning about Galois theory these days. And I am considering to prove:

1. Is that the fact that given a tower of extensions $A/B/C/D$, if $A/B$, $B/C$, $C/D$ are normal, then $A/D$ is normal?

2. Is that the fact that given a tower of extensions $A/B/C/D$, if $A/B$, $B/C$, $C/D$ are separable, then $A/D$ is separable?

If it is true, may I please ask for a proof? Or if it is false, may I please ask for some counterexamples? Any help or reference would be appreciate. Thanks a lot!

Answer 1

Let $A={\bf Q}(\root4\of2)$, $B={\bf Q}(\sqrt2)$, $C=D={\bf Q}$, then $A/B$, $B/C$, $C/D$ are all normal, but $A/D$ isn't.