I'm just musing here, this is only (barely even) a half-formed idea, but I'm just wondering if it's at all a valid train of thought. The proof I read of the insolubility of the quintic polynomial rests on finding a particular quintic which is not solvable. Could a proof of the insolubility of all higher degree polynomials be formed out of the fact that if such a formula existed, say, for degree 6 polynomials, then I could solve an insolvable quintic polynomial by multiplying it by $x$, solving the polynomial I get as a result, then discarding the 0 solution given? Is that a contradiction? Or would that be possible given that there was some formula for higher degree polynomials?
Would it just be a "trick" to solve them?
Surely if one had a formula in the coefficients for a higher degree polynomial, they could then set to zero all coefficients that would become zero and get a formula for solving quintics.