Suppose we have a tower of field extensions:
¯F⊂K⊂E⊂F
Is it true in general that |G(K/F)|=|G(K/E)|⋅|G(E/F)|?
I was able to verify some specific examples, like Q(3√2,ω) for x3−2 and another extension, but how could I show that this holds in general for all such towers of extensions?
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