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.