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See this article from wikipedia: http://en.wikipedia.org/wiki/Bring_radical

George Jerrard showed that some quintic equations can be solved using radicals and Bring radicals, which had been introduced by Erland Bring. They can be used to obtain closed-form solutions of quintic equations.

How is it possible to prove that? Does anyone know an article that gives the proof? I know some basic Galois theory, but I'm not expert at it. Also, can we define ultraradicals for higher powers and obtain closed-form solutions of higher degree equations? If yes, what part of mathematics deal with these problems?

Also, Charles Hermite has a paper where he proves that quintic equations can be solved using elliptic integrals. See here.

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The basic theorem is that a polynomial is solvable by radicals iff its Galois group is solvable. The solvability of the Galois group is the proof you ask for. In this case, the Galois group of the Bring radical is not solvable. So if there were a way to "invert" the Bring radical--i.e. find its roots the same way you find a square root or cube root--then you could solve many more quintics.

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The basic idea is that every quintic can be transformed into the Bring normal form via an elementary change of variables. Thus, when you introduce this one new special function to solve the Bring normal form, you can then unwind the change of variables to get an enormous expression which is a "closed-form" solution to the general quintic, but where "closed-form" now means there are a few terms that involve this new special function.

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