Suppose there exists an intermediate field \(L\) of the Galois extension \(F/E\) of degree 2 over \(E\). What can we say about \(\operatorname{Gal}(F/E)\)?
Solution
1. The subgroup corresponding to \(L\) has index \(2\) in \(\operatorname{Gal}(F/E)\).
Set \[ G=\operatorname{Gal}(F/E) \qquad\text{and}\qquad H=\operatorname{Gal}(F/L). \] By the fundamental theorem of Galois theory, \[ [G:H]=[L:E]=2. \] Thus \(G\) contains a subgroup of index \(2\).
2. Consequently \(H\) is normal and \(G\) has a quotient isomorphic to \(C_2\).
Every subgroup of index \(2\) is normal: the two left cosets and the two right cosets both consist of \(H\) and its complement, so they coincide. Hence \[ H\trianglelefteq G. \] The quotient has order \[ |G/H|=[G:H]=2, \] and therefore \[ G/H\cong C_2. \] Equivalently, \(G\) admits a surjective homomorphism onto \(C_2\). In particular \(|G|\) is even.
This is the general structural conclusion: \(\operatorname{Gal}(F/E)\) has a normal subgroup of index \(2\).