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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:
This formula applies to any algebraic equation of any degree without need for a Tschirnhaus transformation or any other manipulation to bring the equation into a specific normal form, such as the Bring–Jerrard form for the quintic. However, application of this formula in practice is difficult because the relevant hyperelliptic integrals and ...
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 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.
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.
Watson provided proofs for most of Ramanujan's modular equations. Bruce C. Berndt completed the project begun by Watson and Wilson. Much of Berndt's book Ramanujan's Notebooks, Part 3 (1998) is based upon the prior work of Watson. [11] Watson's interests included solvable cases of the quintic equation. He introduced Watson's quintuple product ...
While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as x 5 - 1 = 0, and the precise criterion by which a given quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not ...
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).