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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!

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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.

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  • $\begingroup$ Got it. And if possible, may I please ask if the second point holds? $\endgroup$ – PropositionX May 26 '17 at 9:30
  • $\begingroup$ I think so. See Galois Theory by Stewart. $\endgroup$ – Gerry Myerson May 26 '17 at 12:37
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    $\begingroup$ Yes, a tower of separable extensions is separable. This follows from the multiplicativity of the “separable degree”, if you know about that. The difficulty of the whole matter depends strongly on what your definition of separability is. $\endgroup$ – Lubin May 28 '17 at 23:24

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