I can get my head around this so someone explain it please.
$(1)$ From Galois theory it is known there is no formula to solve a general quintic equation.
But it is known a general quintic can be solved for the 5 roots exactly. Back in 1858 Hermite and Kronecker independently showed the quintic can be exactly solved for (using elliptic modular function). Also I think they're maybe other solution for the quintic which means a formula for each of the 5 roots.
So why is the claim in Galois theory that there is no formula to solve it?
I know I am missing something here because the above $(1)$ is an established result.
So what is the value in saying, using Galois theory we do not have a formula for the 5 roots? Since for practical purposes we can actually find the 5 roots each time using say for example the formula based on elliptic modular functions.
Best Answer
Galois-theory only says that there is no general formula to solve a quintic equation in terms of radicals. That is, there is no formula only using the arithmetic operations "sum, multiplication etc. and taking the $n$-th root".
For instance for the polynomial $x^5 - 4x + 2$ it is known that it has a root that is not expressible in the above mentioned operations (as its Galois-group is $S_5$). (Edit: Another example is $x^5 + x + 1$ which also has Galois-group $S_5$. If you Wolframalpha this polynomial you see nicely how four of its roots can be expressed by radicals, but the fifth can't.)
The solution you mean is the solution using Bring radicals - wikipedia-article here: https://en.wikipedia.org/wiki/Bring_radical -, which is not a contradiction, as it is not expressed in form of radicals (in the sense of $n$-th roots of something).