Proof in the context of Galois theory that quintics and above can't be solved by the trigonometric and exponential function

I have been studying Galois theory as of late and recently I came across this proof in the context of Complex Analysis which not only prove the unsolvability of the general quintic and higher degree polynomials using radicals but also using trigonometric and exponential functions.

Is there a proof in the context of Abstract Algebra/ Galois theory that is equivalent? Preferably this proof should follow the same line of reasoning as the Abel-Ruffini theorem (the version that I found on Wikipedia specifically).