By Galois Theory an arbitrary polynomial equation of degree $\ge 5$ is not solvable using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the alternating group of order 5, the symmetry group is not soluble).
Is this the only way an arbitrary polynomial equation can be (exactly) solved?
Does Galois Theory provide universal concept of general solvability?
Are there other methods except using radicals (for exact, not approximate or numerical solution)?
PS. A related question
Best Answer
As my Group theory professor told me, giving a root of $x^2-2=0$ the name $\sqrt 2$ doesn't magically "solve" the equation. Calling a number $\sqrt 2$ is just saying "it solves $x^2-2=0$". You haven't gained any new information. Similarly, if $p(x)$ is an irreducible polynomial, then simply writing down $p(x)=0$ could be considered to be "solving" the equation, in the sense that, in a certain technical sense, the fact that $x$ solves that equation contains all relevant information about $x$.
Solving by radicals means that we wish to express roots of a polynomial in terms of the roots of a particular family of polynomials $x^2-a$. One reason to do this is that it means that we can numerically approximate those roots using methods to numerically approximate roots of that simpler family of polynomials. This is a common strategy in mathematics. For example, in trigonometry we try to express all quantities using $\cos$ and $\sin$. In a sense, this doesn't mean we've "calculated" those quantities, since $\cos$ and $\sin$ are just names given to particular ratios of line segments, but it does mean that we can reduce the problem of approximating various quantities to just approximating $\cos$ and $\sin$.
The Bring radical represents the applicaiton of this strategy to quintics. We choose a particular one-dimensional (in the vector space sense) family of polynomials and invent a special symbol to refer to a particular root of a polynomial in that family. It can be shown that in this way we can express the roots of any quintic.