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Proof of Algebraic Solution of the General Quintic Equation, Overlooked Dimensions in Abel-Ruffini Theorem

Samuel Bonaya Buya

I prove that the general quintic equation is solvable in radicals. Here solvability in radicals is loosely used to mean root form of the quintic equation even though there may be cases formula for the equation may result in a rational root. I re-examine Abel-Ruffini theorem in the light of Galois Theory in order to highlight some avenues possibilities and loopholes not thoroughly explored in arriving at the theorem. I show that the general quintic equation has a solvable Galois group. Proof of solvability in radicals comes down to computing the Galois group of the general quintic equation and showing that it is solvable. Since S5 is the Galois group of the quintic equation I will show and demonstrate that the quantities used to construct it are as a matter of fact algebraically determinate. In the process of proving this I will derive all the roots of the general quintic equation. In so doing I prove the incompleteness of Abel-Ruffini theorem and Galois Theory at large. In attempting to prove the solvability of the general quintic in radicals, a very important principle is discovered governing solution of polynomial equations