I am browsing through the book "Abel’s Theorem in Problems and Solutions" which outlines a topological proof of the Abel-Ruffini theorem (due to V.Arnold).
For a polynomial $p$, one investigates the monodromy groups of the Riemann surface of $\omega(z)$, where $p(w(z))+z=0$. Theorem 11 in section 2.13 then says:
If the multi-valued function $\omega(z)$ is representable by radicals, its monodromy group is solvable.
Remark 1 in the corresponding section says that one is also allowed to use, besides radicals, for example all analytic single-valued functions ($e.g. \sin{z}$) and also $\log{z}$. Unfortunately, I find no further explanation in the book of how this works. Is there a reference where this is outlined in more detail?
In this setting, as indicated in this post, there is a connection/isomorphism bewteen the monodromy group and Galois groups. Can one conlude with this "isomorphism" that: If the Galois group is not solvable, the roots of $p(x)$ are not expressable in radicals and trigonometric functions?
Let $F=\Bbb{C}(z)$ and a polynomial $g\in F[X]$, factorize $g(X)=\prod_j (X-w_j)$ and $K=F(w_1,\ldots,w_d)$ its splitting field.
You are looking at $g=p(X)+z$ with $p\in \Bbb{C}[X]$.
Radical means that $w_j$ has an expression which is a tree of additions, multiplications, quotients and $n$-th roots of elements of $F$.
In every case, away from the finitely many poles and branch points located at the zeros/poles of the coefficients of $g$ and $Disc(g)\in F$ then $w_j$ is an analytic function of $z$. When $g$ is not solvable then this analytic function has no simple expression (this happens for most polynomial of degree $\ge 5$).
Next, $Gal(K/F)$ is the monodromy group of $w_1$. Take a base point $z_0\in \Bbb{C}$ where all the $w_j=w_j(z)$ are analytic, then $\gamma \in Gal(K/F)$ can be represented as a closed curve $z_0\to z_0$ and $\gamma(w_j)$ is the function analytic near $z_0$ obtained by continuing $w_j$ analytically along $\gamma$.
This is because if you take $u_1,\ldots,u_M$ all the functions analytic near $z_0$ obtained by analytic continuation of $w_1$, then (since the analytic continuation commutes with the algebraic operations) all of them are roots of $g$, and $\prod_{m=1}^M (X-u_m(z))$ is a polynomial whose coefficients are locally analytic away from finitely many poles and branch points, and those coefficients stay the same under analytic continuation along closed-loops. Together with the polynomial growth at $\infty$ and near the branch points this implies that the coefficients of $\prod_{m=1}^M (X-u_m(z))$ are meromorphic on the Riemann sphere: they are in $F$. Whence $g=\prod_{m=1}^M (X-u_m(z))$.