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Solving quintic equations in terms of radicals (nth roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.
The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form: + + + + + = The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using Tschirnhaus transformations to reduce the number of independent coefficients.
Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals.
Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equations. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable [1] (note this theorem holds only in characteristic 0).
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Something more general is required for equations of higher degree, so to solve the quintic, Hermite, et al. replaced the exponential by an elliptic modular function and the integral (logarithm) by an elliptic integral. Kronecker believed that this was a special case of a still more general method. [1]
This is a cubic equation in y. Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.
George Paxton Young (9 November 1818 – 26 February 1889) was a Canadian philosopher and professor of logic, metaphysics and ethics at the University of Toronto. [1] [2] He studied the quintic polynomial equation and in 1888 described how to solve a solvable quintic equation, without providing an explicit formula.