At the moment I am reading Chapter 32 in Gallian's book "Contemporary Abstract Algebra", ninth edition. I am stuck with part of the proof of the following Theorem.
"Let F be a field of characteristic 0 and let f(x) $\in$ F[x]. Suppose that f(x) splits in F($a_1, ...,a_t$), where $a_1^{n_1} \in F$, $a_i^{n_i}\in F(a_1,..,a_{i-1})$ for i=2,...,t. Let E be the splitting field of f(x) over F in F($a_1, ...,a_t$) . Then Gal (E/F) is solvable."
The proof is by induction on t. I understand the proof that the theorem holds for t=1. But I am stuck with the induction step.
The second part of the proof starts like this.
"Now suppose t>1. Let a = $a_1^{n_1} \in F$, let L be a splitting field of $x ^ {n_1} - a$ over E and let $K \subset L$ be the splitting field of $x ^ {n_1} - a$ over F."
This is where I am stuck. I cannot see how these fields are related. Do we have $F \subset E \subset K \subset L$? Or are they otherwise related?
I hope someone can point me in the right direction. I have the feeling that with a basic idea of what is going on here the rest of the proof will become clear.
$E$ and $K$ are both intermediate extensions of $L/F$, i.e. $F \subset E \subset L$ and $F \subset K \subset L$. Furthermore, $L = KE$, i.e. $L$ is the compositum of $K$ and $E$ over $F$.
The big picture of this in the proof is that instead of focusing on the extension $E/F$, we are transfering the situation to $L/K = EK/K$ instead, i.e. we are "changing the base" from $F$ to $K$.
All the intermediate steps in $E/F$ (i.e. adding the roots one by one) are all copied to become intermediate steps in $L/K$.