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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:
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]
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.
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 conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r , then the L -function L ( E , s ) associated with it vanishes to order r at s = 1 .
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
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
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).