Abel and Galois (and Arnold)

The action of the monodromy group of $w(z)$ on the fiber $p^{-1}(a)$ for a non-critical value $a$ of $p$ (that is $|p^{-1}(a)|=\deg p$) is the same as the action of the Galois group of $p(x)+z$ over $\mathbb C(z)$ on the roots of $p(x)+z$ in some splitting field. One can see this by comparing each of these groups with the deck transformation group of the cover $X\to\mathbb P^1\mathbb C$, where $X$ is the normal hull of the cover $P^1\mathbb C\to P^1\mathbb C$, $x\mapsto p(x)$. (Remark: Though it doesn't make a difference, it is slightly more convenient to work with $p(x)-z$ instead of $p(x)+z$.)

Yes, this was explored at length in the work Joseph Fels Ritt in the 1920s-1930s, who wrote it all up rather well, but it also appears in various Serre books (Topics in Galois Theory, I am pretty sure).

I am sure it appears in Serre's books, and certainly Topics in Galois Theory talks a lot of the function field case. As for Ritt, here is a random example (he wrote a lot on this in the twenties, most quite relevant):

Ritt, J. F., On algebraic functions which can be expressed in terms of radicals., American M. S. Trans. 24, 21-30 (1923). ZBL49.0717.01.

Another good source from all this is Jean-Pierre Tignol's book:

Tignol, Jean-Pierre, Galois’ theory of algebraic equations, Singapore: World Scientific. xiii, 333 p. (2001). ZBL0972.12001.