History. First solved by Abel / Galois in the 1800s. Our solution follows Arnold's topological solution from the 1960s. I could not find the original writeup by Arnold (if it at all exists), yet see:
V.B. Alekseev, Abel's Theorem in Problems and Solutions, Based on the Lecture of Professor V.I. Arnold, Kluwer 2004.
A. Khovanskii, Topological Galois Theory, Solvability and Unsolvability of Equations in Finite Terms, Springer 2014.
B. Katz, Short Proof of Abel's Theorem that 5th Degree Polynomial Equations Cannot be Solved, YouTube video, http://youtu.be/RhpVSV6iCko.