I've been trying for a long time to fully understand the definition of solvability by radicals of a polynomial. These are the two definitions I have.
Definition 1: We say that a field extension $F/K$ is a radical extension if we can form a chain of fields
$$K=K_0 \leq K_1 \leq \cdots \leq K_n=F$$
where $K_{i+1}/K_i$ is a simple extension such that $K_{i+1}=K_i(a_i)$ and $a_i^{k_i} \in K_i$ for some positive integer $k_i$.
Definition 2: Let $K$ be a field and $f(x) \in K[x]$, we say that $f$ is solvable by radicals if there exits a radical extension $F/K$ such that $F$ contains a splitting field of $f$ over $K$.
Now, I know that definition 2 formalize the concept that we can express the roots of a polynomial as some formulas that only involves the basic operations on the field over the coefficients of the polynomial (and some fixed elements of the field) and taking roots of that elements.
I can see why if we have such formulas for the roots of a polynomial we can interatively construct for every root a radical extension that contains that root, and we can then "join" them to get a radical extension that contains the spplitting field of the polynomial.
But, the reciprocal of this reasoning is what I'm not really able to see. I can see that if we have a polynomial $f$ and it is solvable by radicals, then we have a radical extension $F/K$ containing all roots of $f$. Then, as we can iteratively express $F=K(a_0, \cdots, a_{n-1})$ where every $a_i$ is in a way a "nested chain of roots" of elements of $K$, we conclude that as every root of $f$ is in $F$, we can express it as sums and products of elements of $K$ and "chains" of roots of some more elements of $K$.
Now, my concrete doubt, althoug I was able to get an expresion for the roots of the polynomial in a "radical formula", the coefficients of the polynomial doesn't play any role here. Is there a way to show that every root of a solvable by radicals polynomial is expresable in the form of a "radical formula", as the ones mentioned, but involving explicitly the coefficients of the polynomial?
I've been searching for this in a lot of references but I haven't found anything that explicitely discuss this, so I'll be really greatful to anyone who can help me with this.
Thank you very much.