I was ruminating over quintics and got curious about the following idea.
Consider a quintic equation:
$$ Q(x) a_0 + a_1 x + a_2 x^2 + ... a_5 x^5$$
Such that the solutions to
$$ Q(x) = 0 $$
Are expressible by radicals. Then is there a general purpose formula $f(a_0, a_1 ... a_5)$ that can be used to express them?
This seemed to be relevant: https://en.wikipedia.org/wiki/Quintic_function#Quintics_in_Bring.E2.80.93Jerrard_form
But it's not obvious to me after reading that, that ANY radical-solvable quintic, can be properly depressed through the mentioned transformations, and then solved using that formula.
This has me curious as i'm considering the alternative application:
Given a quintic that one wants to test is solvable, instead of computing its galois group, one could instead just use the formula to come up with "potential solutions" and then algebraically verify whether they work, to conclude if it is solvable or not.
Daniel Lazard implemented a program in Maple that solves for quintics that are solvable by radicals: you can take a look at this book published by Springer (2002): The Legacy of Niels-Henrik Abel