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For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial equation $p(x) = 0$, when the degree is $\geq 5$ there isn't a solution in this form with the help of radicals due to an application of Galois theory.

My question: how far can you get with this kind of (a,b,c)-formulae when you allow non-radical solutions when solving polynomial equations ?

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    Are you familiar with solutions obtained using Bring radicals, Jacobi theat functions, and elliptics modular functions? See e.g. [Wikipedia, beyond radicals](http://en.wikipedia.org/wiki/Quintic_function#Beyond_radicals)2012-01-04
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    possible duplicate of [Finding roots of polynomials with rational coefficients](http://math.stackexchange.com/questions/68178/finding-roots-of-polynomials-with-rational-coefficients)2012-01-04
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    In any event: [Umemura](http://books.google.com/books?id=jPFaLY31gwYC&pg=SA3-PA261) shows how to use [Riemann theta functions](http://dlmf.nist.gov/21) to represent roots of arbitrary-order polynomials. See [Glasser's preprint](http://arxiv.org/abs/math/9411224) as well.2012-01-05

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