Problem Related to an Earlier Question

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The following question has been already asked on this site before here- Proposition 11 (Chapter 1, Section 5) in Lang's Algebraic Number Theory. I now want to generalize this statement for infinite Galois extensions using Zorn's Lemma. Can anybody please help me with this? As I have no exposure to infinite Galois Theory, a detailed proof will be most helpful in this case. Thanks for any help in advance.