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It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic closed-form formulas. And I remember reading somewhere that the roots of quintic equations can be expressed in terms of the hypergeometric function.

What is known, beyond Abel-Ruffini, about closed-form formulas for roots of polynomials? Does there exist a formula if we allow the use of additional special functions?

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Transcendent functions do the job sometimes. For example, the roots of $X^n - 1$ are given by $e^{2 \pi k / n}$. – Hans Giebenrath Feb 1 '13 at 6:50
@HansGiebenrath, roots of $X^n -1$ count as obtained by extraction of radicals. – Andreas Caranti Feb 1 '13 at 7:22
Indeed the Galois group of $X^n-1$ is not only solvable, but also abelian. – JSchlather Feb 1 '13 at 7:26
I'm sorry. You are right. I was aiming at CM-fields and extensions thereof generated by values of $j$. I think it should be possible to construct a non-solvable extension generated by the value of $j$. – Hans Giebenrath Feb 1 '13 at 9:32

3 Answers 3

As you might already know, solutions to the quintic can be expressed in terms of either ${}_4 F_3$ hypergeometric functions or Jacobi theta functions. See King or Prasolov/Solovyev for details.

For polynomials of higher degree, there is also a general formula for the roots, due to Umemura. The formulae involve the multidimensional generalization of the Jacobi theta functions (the Riemann theta function), and are a bit unwieldy; see Umemura's paper if you want more details. See also this preprint for a solution of the reduced polynomial equation $x^n-x-\alpha=0$ in terms of hypergeometric functions.

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The question is based on

general polynomial of degree 5 or higher have "no closed-form formula"

This is not exactly true, what it should be said is that "general algebraic equations of degree higher than 4 do not admit solutions by radicals" which means that they cannot be solved by operations implying combinations of ordinary additions, multiplications, divisions, raising to powers, root extractions... On the other side it was shown by Hermite that fifth degree equations can be solved using the modular elliptic functions, which provide a generalization of the so called trigonometric solution of eqs. with degree lower than 5. Higher order (than 5) algebraic equations can be solved by employing other forms of elliptic functions.

In more recent times the use of the Lagrange inversion formula has allowed solutions in terms of hypergeometric functions. This technique was developed by an italian mathematician G. Belardinelli in 1959 and later rediscovered by M. L. Glasser in 2000 J. Comp. Appl. Math. 118 (2000) 169-171.

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Doesn't the lagrange inversion formula give you a 'formula' for the roots? (as an expansion from some point $a$ nearby)

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