If is a Galois extension, , and the intermediate fields are both degree 2 over , then .
Using the tower formula, we know which means that . Since are intermediate fields, they must be Galois extensions. By the Fundamental Theorem of Galois Theory, , so both and are cyclic, isomorphic to , and hence normal subgroups. But
, 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.