Notes on Galois Theory III
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چکیده
Clearly F ≤ EH ≤ E. On the other hand, given an intermediate field K between F and E, i.e. a subfield of E containing F , so that F ≤ K ≤ E, we can define Gal(E/K) and Gal(E/K) is clearly a subgroup of Gal(E/F ), since if σ(a) = a for all a ∈ K, then σ(a) = a for all a ∈ F . Thus we have two constructions: one associates an intermediate field to a subgroup of Gal(E/F ), and the other associates a subgroup of Gal(E/F ) to an intermediate field. In general, there is not much that we can say about these two constructions. But if E is a Galois extension of F , they turn out to set up a one-to-one correspondence between subgroups of Gal(E/F ) and intermediate fields K between F and E, i.e. fields K with F ≤ K ≤ E.
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