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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.
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.
which expresses the solutions of the quadratic equation + + = There exist algebraic solutions for cubic equations [1] and quartic equations, [2] which are more complicated than the quadratic formula. The Abel–Ruffini theorem, [3]: 211 and, more generally Galois theory, state that some quintic equations, such as
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]
Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree. The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation.
The general quintic may be reduced into what is known as the principal quintic form, with the quartic and cubic terms removed: + + + =. If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation = + +, the coefficients and may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities.
This is a list of equations, ... Biquadratic equation; Quartic equation; Quintic equation; Sextic equation; Characteristic equation; Class equation; Comparametric ...
The general quintic equation in Bring-Jerrard form: x 5 − 5 x − 4 a = 0 {\displaystyle x^{5}-5x-4a=0} for every real value a > 1 {\displaystyle a>1} can be solved in terms of Rogers-Ramanujan continued fraction R ( q ) {\displaystyle R(q)} and the elliptic nome