For a field $F$ such that $char(F)\ne 2$. Consider the tower of extension $E/L/F$ where $L = F(√c)$ and $E = L(\sqrt{a + b\sqrt{c}})$, for some $a,b,c ∈ F$.
Suppose we have $E/F$ is not Galois, I need to prove that $M=E(\sqrt{a-b\sqrt{c}})$ is the minimal Galois extension of $F$ .
Here minimal is in the sense that no smaller subfield of $M$ containing $E$ is Galois over $F$.
I have spent some time on the question, so I would really appreciate if someone can give me some explaination. Thanks a lot.
Also I think I need a picture of extensions and correspondence to see it cleae, an reference would also be appreciate, thanks!