Classify the Galois group of tower of degree $2$ extension.

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 $a,b,c ∈ F$.

When the degree-$4$ extension is Galois, it is said that there are only two possibilities for its Galois group, up to isomorphism. May I please ask for what is the $2$ possiblities? And under what condition on $a,b,c$?

And also, for the case which I am considering, may I please ask for an explicit explaination that what elements generate the subextensions of $E/F$? Could someone please give me a picture of the extensions so that would be made clear.

I know that it seems to be quite a general stuff... I think I do need some explaination. Thanks for help or reference!

• Using Galois correspondence, the Galois group of the degree $4$ extension has order $4$. There are only two groups of order $4$ (up to isomorphism): cyclic group of order $4$ and Klein-$4$. – Alex Vong May 23 '17 at 14:12
• @AlexVong Yes I did notice that as the degree of extension is $4$ we must have a Galois group of order 4. So may I please ask how to base on the situition of $a,b,c$, classify what condition lead to which of these 2 groups? – PropositionX May 23 '17 at 14:19
• Have a look at this post math.stackexchange.com/q/649466/254075 – sharding4 May 23 '17 at 14:24
• He says that a sufficient condition for the splitting field of $P(x) = (x^2-\alpha^2)(x^2-\beta^2)$ irreducible having Galois group $\mathbb{Z}_4$ is $\alpha\beta(\alpha-\beta) \in F$ – reuns May 23 '17 at 14:51
• @sharding4 Thanks for the reference! I'll have a look. – PropositionX May 23 '17 at 14:54