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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, 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 – Bill Dubuque Jan 4 '12 at 22:26
possible duplicate of Finding roots of polynomials with rational coefficients – J. M. Jan 4 '12 at 23:56
In any event: Umemura shows how to use Riemann theta functions to represent roots of arbitrary-order polynomials. See Glasser's preprint as well. – J. M. Jan 5 '12 at 0:02

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