# Clarification of Abel-Ruffini theorem statement

From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations.

How does it imply that we can't have a formula that involves some other mathematics thus allowing us to find all the roots of any polynomial? Do abstract algebra and Galois theory have a role in this I'm not aware of?

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It does not, actually. We can construct solutions of quintics using mathematical function, yes. Lame way to do this : $f(a)$ where $f(x)$ is defined to be the inverse function of $x^5 + x$. This is the "so-called" bring radical. Smart way to do this : Elliptic functions. Explicitly given here. –  Balarka Sen May 9 at 19:48