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Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form:
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]
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.
The technique I presented due to M.L. Glasser has been generalized to equations of arbitrarily high degree but using hypergeometric functions of several variables (see here for a German paper). The original quintic solution by Charles Hermite was later generalized to equations of arbitrary degree using Siegel modular forms.
The general case of quintic equations in the Bring–Jerrard form has a non-elementary solution based on the Abel–Ruffini theorem and will now be explained using the elliptic nome of the corresponding modulus, described by the lemniscate elliptic functions in a simplified way.
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).