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

Using the tower formula, we know $\displaystyle 10 = [E:F] = [E:K][K:F] = n \cdot 2$ which means that $\displaystyle [E:K] = 5 = [E:L]$. Since $\displaystyle K,L$ are intermediate fields, they must be Galois extensions. By the Fundamental Theorem of Galois Theory, $\displaystyle |\text{Gal}(E/K)| = 5 =|\text{Gal}(E/L)|$ , so both $\displaystyle \text{Gal}(E/K) $ and $\displaystyle \text{Gal}(E/L) $ are cyclic, isomorphic to $\displaystyle \mathbb Z / 5 \mathbb Z$ , and hence normal subgroups. But

$\displaystyle \text{Gal}(E/K) \cong \text{Gal}(E/L)$ , 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.