Suppose I have fields F≤L1<L2≤E, where E/F is separable, [E:F]=pn for a prime number p, and L2/L1 is Galois and of degree p. In addition, I have a subfield F≤K≤E. Is it true then that L2∩K/L1∩K is Galois (and of degree 1 or p)?
I have proven this in the analogous case for groups, where you replace Galois-ness with normality. However, that proof relies on the Second Isomorphism Theorem, and I cannot find a way of adapting it to the case of fields. As a partial result, E/F is separable, so L2∩K/L1∩K is separable as well, so we only need to show it is also normal to get Galois.
Edit: This is my attempt at solving a more general question I asked here, so it may turn out that this statement is false if this isn't the correct approach.
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