closed-form expression for roots of a polynomial

Question

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?

Answer 1

Polynomial equations can be solved with the 4 arithmetic operations, prime power roots and a suitable collection of other special roots obtained from a subset of the "unsolvable" equations of 5 and higher order. For the quintic equation, if I recall correctly, one needs only add the "star-root" x = *√c to x⁵ + x = c of real numbers c to the list of operations.

That's described in passing in R. Bruce King, Beyond the Quartic Equation, Birkhäuser, Boston, Basel, Berlin, 1996.

Answer 2

You are right: "no closed-form formula" is not the right term here, because "closed form" means allowed constants, functions and operations from a given set.

Here are general closed-form solution formulas for quintics:
https://en.wikipedia.org/wiki/Bring_radical#Solution_of_the_general_quintic
How to solve fifth-degree equations by elliptic functions?

I have an addition to the other answers and the generally known:
The explicit elementary numbers are generated from the rational numbers by applying finite numbers of $\exp$, $\ln$ and/or radicals. Clearly, the elementary numbers contain all algebraic numbers and the explicit elementary numbers contain all explicit algebraic numbers (the numbers representable by radicals).
Chow [Chow 1999] gives his Corollary 1:
"If Schanuel's conjecture is true, then the algebraic numbers in" the explicit elementary numbers "are precisely the roots of polynomial equations with integer coefficients that are solvable in radicals."
That means, the algebraic equations that cannot be solved by radicals cannot be solved by elementary numbers (means by applying elementary functions).
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[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448