See e.g.:

Vladimir Arnold: Collected Works. Springer

Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014

Khovanskii, A.: One dimensional topological Galois theory. 2019

Askold Khovanskii - Publications

For the topological proofs without connection to representability in meromorphic functions, see e.g.:

 https://math.stackexchange.com/questions/1550990/is-there-an-alternative-proof-of-the-abel-ruffini-theorem

See e.g.:

• Vladimir Arnold: Collected Works. Springer

• Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014

• Khovanskii, A.: One dimensional topological Galois theory. 2019

• Askold Khovanskii - Publications

For the topological proofs without connection to representability in meromorphic functions, see e.g.:  Is there an alternative proof of the Abel–Ruffini theorem?.