My Galois theory book has a problem on resolvents where you're supposed to prove that you can solve the quintic: x5 + px + q = 0 by radicals over the field that contains one of the roots of the resolvent polynomial and the square root of the discriminant.
The resolvent polynomial is sixth order, and there's no explicit formula for a sixth order by radicals any more than a fifth order, so it's not like a useful way to do it or anything. Plus, it sounds like there's some fancy method using hypergeometric functions if you're after something explicit.
To do the exercise, you were just supposed to show the corresponding group is solvable, but I came up with an explicit formula for it too. I have no idea if anyone has done it before, and I don't really know why they would have. I guess calling it a proof is a little weird because it involved solving these big matrix equations on my computer, but it was a fun project.
The final formula is huge and I only ever really computed it in a python script, but it does give the same answers as the polyroots function. If you're interested: https://pastebin.com/BGwDEjbR
With those values of p and q, I got: p=-4.200000 q=-8.300000 permuting the polyroots with: [4, 2, 1, 0, 3] error: 7.622009e-14 polyroots analytic +1.73171 + j+0.00000 vs +1.73171 + j-0.00000 +0.31186 + j-1.60664 vs +0.31186 + j-1.60664 -1.17771 + j+0.63432 vs -1.17771 + j+0.63432 -1.17771 + j-0.63432 vs -1.17771 + j-0.63432 +0.31186 + j+1.60664 vs +0.31186 + j+1.60664