TITLE: Abel-Ruffini theorem, Galois theory and minima and maxima
QUESTION [11 upvotes]: Questions:

Does there exist a proof of the Abel-Ruffini theorem without using Galois theory?
Does there exist a proof that there exists a polynomial $P$ with $\deg P = 5$ such that the roots are not expressible in radicals? Can this be proven without Abel-Ruffini?
Does there exist a polynomial $P$ with $\deg P = 6$ such that the minima and maxima are not expressible in radicals? Can this be proven without Abel-Ruffini?


I originally had the following proof for the third question:

The minima and maxima of a 6th degree polynomial are the roots of its derivative, which is a fifth degree polynomial. There exists a 5th degree polynomial such that the roots are not expressible in radicals, otherwise the roots of every 5th degree polynomial would be expressible in radicals, contradicting the more general form of the Abel-Ruffini theorem proven by Galois. Therefore there exists a 6th degree polynomial whose minima and maxima are not expressible in radicals.

However, I realized only until later, that it also can be done vice versa. However, the question brought up the several other questions which I listed on top of the question. Therefore I have clarified those questions up in the answer below.

REPLY [3 votes]: Q1. Yes. Abel's original proof is an example. Another example is the proof provided by Arnold, which makes use of group theory and topology.  
Q2. This implies the Abel-Ruffini theorem since if there exists a polynomial $P$ with $\deg P = 5$ such that the roots are not expressible in radicals there is certainly no general formula that gives the roots. Note that Abel-Ruffini doesn't imply this. In fact the result is by Galois. 
Q3. If there exists a polynomial $P$ with $\deg P = 6$ such that the minima and maxima are not expressible in radicals, then the roots of the derivative, a fifth degree polynomial, are not expressible in radicals. By the result in Q2, we conclude this also implies the Abel-Ruffini theorem.