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I am trying to understand the original proof of Abel-ruffini for the insolvability theorem of quintic equation.
I can not follow the logic in the paper at many steps.
I will ask my doubts in different posts (if that is ok to moderators). It is written in the paper that if $$AY^5+BY^4+CY^3+DY^2+EY+F=0......(2)$$
'is solvable by radicals then,
$$Y=a+bR^{1/5}+cR^{2/5}+dR^{3/5}+eR^{4/5}.......(1)$$
He proves it by saying that $$v=F/G = f+g^{1/p}+h^{2/p}+i^{3/p}+j^{4/p}$$ I follow that F/G can be expressed in that way (always), but
$1.$how does that prove his claim in ($1$)? (Here R is an algebraic function.)

2.He then substitutes $(1)$ in $(2)$ and gets$$k+lR^{1/p}+mR^{2/p}+nR^{3/p}+oR^{4/p}=0$$ his next claim is that $$l=m=n=o=0$$ How ?
3.Abel observes that if $y$0 is a solution to $$y^5-ay^4+by^3-cY^2+dy-e=0$$
so are the values of $y$i where $y$i are obtained by multiplying $y$0 with 5th root of unity. How?

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