In an assignment I got I have been asked to try to determine when $E/F$ is a Galois Extension, and determine the Galois group of such an extension.
$\textbf{Context:}$ $F$ is any field of characteristic zero and $E = F(\sqrt{c},\sqrt{a + b\sqrt{c}})$, where $a,b,c \in F$, $c$ is not a square in $F$ and $a + b\sqrt{c}$ is not a square in $F(\sqrt{c})$. Now assume that $b \neq 0$; I have determined that this is a Galois extension iff
$\sqrt{ a -b\sqrt{c}}$ is in $E$. This holds iff $a - b\sqrt{c}$ is the square of an element in $F(\sqrt{c})$, or $(a + b\sqrt{c})(a - b\sqrt{c} ) = a^2 - b^2c$ is the square of an element in $F(\sqrt{c})$. Now the first case cannot hold because it contradicts our assumption that $a + b\sqrt{c}$ is not a square in $F(\sqrt{c})$. So the second case holds, that is when $a^2 -b^2c = (h + g\sqrt{c})^2$ for some $h,g \in F$.
Expanding this out and comparing coefficients, it must be the case that either $h = 0$ or (exclusively) $g = 0$. The first case gives that
$a^2 - b^2c = g^2c$ for some $g \in F$, or (exclusively)
$a^2 - b^2c = h^2$ for some $h \in F$.
$\textbf{Where I'm stuck:}$ Now the problem for me comes in determining the Galois group of $E/F$. Firstly $E$ can be written as $F(\sqrt{a + b\sqrt{c}}, \sqrt{a - b\sqrt{c}})$ or even just $F(\sqrt{a + b\sqrt{c}})$.
If we take $E = F(\sqrt{a + b\sqrt{c}}, \sqrt{a - b\sqrt{c}})$ then noticing that $E$ is the splitting field of $(x^2 - a)^2 - b^2c$ over $F$, my guess is that $\sigma \in \operatorname{Gal}(E/F)$ can only take $\sqrt{a + b\sqrt{c}}$ to $- \sqrt{a + b\sqrt{c}}$, it can't take $\sqrt{a + b\sqrt{c}}$ to say $\sqrt{a - b\sqrt{c}}$. I am guessing this based on looking at the polynomial $(x^2 - a)^2 - b^2c$ which can be written as
$(x^2 - (a + b\sqrt{c}))(x^2 - (a - b\sqrt{c})).$
However it seems to me that I am not taking advantage of the conditions found for $a^2 - b^2c$ above. In addition how can I determine whether given some $\sigma$ a permutation of the roots, that it is actually a valid member of the Galois group $\operatorname{Gal}(E/F)$?
Thanks.
$\textbf{Edit:}$ I think the second case where $a^2 -b^2c = h^2$ cannot hold because this would contradict the fact that the degree of $E = F(\sqrt{a +b\sqrt{c}},\sqrt{a - b\sqrt{c}})$ over $F$ is 4.