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    Jes
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    Isomorphism of Galois extensions

    If E/F is a Galois extension, [E:F] = 10, and the intermediate fields K,L are both degree 2 over F, then K \cong L.

    Using the tower formula, we know 10 = [E:F] = [E:K][K:F] = n \cdot 2 which means that [E:K] = 5 = [E:L]. Since K,L are intermediate fields, they must be Galois extensions. By the Fundamental Theorem of Galois Theory,  |\text{Gal}(E/K)| = 5 =|\text{Gal}(E/L)| , so both \text{Gal}(E/K) and  \text{Gal}(E/L) are cyclic, isomorphic to  \mathbb Z / 5 \mathbb Z , and hence normal subgroups. But
     \text{Gal}(E/K) \cong \text{Gal}(E/L) , so I'm thinking this is enough to conclude that the intermediate fields are isomorphic, but I don't know for sure since I can't find anything that says isomorphic Galois groups correspond to isomorphic fixed fields. Thanks.
    Last edited by Jes; February 9th 2011 at 10:05 PM. Reason: still typing
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