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I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general polynomial of degree greater than four.

When I typed such a polynomial into wolfram alpha, it gave me decimal approximations of roots, and that was the only type of expression there was.

My question is, do we have any idea what how we can express a general root other than via decimal approximations?

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Danny Macovei
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    We can speak of "the largest real root of quintic so-and-so" just as we introduce $\sqrt a$ as a symbol for "the largest real root of $X^2-a$" – Hagen von Eitzen Apr 16 '16 at 20:24
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    You could use Bring radicals for a general formula to quintic roots of $x^5+px+q$. – Noble Mushtak Apr 16 '16 at 20:44
  • see also: https://math.stackexchange.com/questions/1555743/how-do-you-solve-5th-degree-polynomials/4229237#4229237 https://math.stackexchange.com/questions/1828551/polynomials-with-degree-5-solvable-in-elementary-functions/4225322#4225322 https://math.stackexchange.com/questions/1068726/when-does-a-formula-for-the-roots-of-a-polynomial-exist https://math.stackexchange.com/questions/291909/closed-form-expression-for-roots-of-a-polynomial – IV_ Aug 21 '21 at 15:27

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You ask for a general method other than via decimal approximation. Because of Galois, such a method must be transcendental, as opposed to algebraic (which you call "arithmetic"). For instance, by analyzing the symmetries of the icosahedron, Felix Klein showed that the solutions of the quintic could be expressed by taking radicals of values of hypergeometric functions (the formulae are analogous to those giving the algebraic solutions of the cubic). A variant is explained in every detail in the memoir "Résolution de l'équation du cinquième degré" by P. Gabriel & S. Benzekry, www.ann.jussieu.fr/~gabriel/documents/Memoire.pdf. The main steps are:
- by using the Newton relations between the coefficients and the roots, transform the general quintic into a "principal form" $X^5 + 5aX^2 + 5bX + c$
- by using the symmetries of a polyhedron inscribed in the Riemann sphere, transform this into the so called Brioschi equation, which is another particular quintic
- transform this in turn into the Jacobi equation, which is a sextic whose coefficients involve the classical Weierstrass functions \Delta and $g_2$
- solve the Jacobi equation by bringing it back to a cubic

nguyen quang do
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