Quintic function

Graph of a polynomial of degree 5, with 4 critical points.

In mathematics, a quintic equation is a polynomial equation of degree five. It is of the form:

ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0\,,

where $a,b,c,d,e,f$ are members of a field, (typically the rational numbers, the real numbers or the complex numbers), and $a\neq 0$ .

Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a quintic function is a quartic function.

Finding roots of a quintic equation

Finding the roots of a polynomial — values of $x$ which satisfy such an equation — in the rational case given its coefficients has been a prominent mathematical problem.

Solving linear, quadratic, cubic and quartic equations by factorization into radicals is fairly straightforward when the roots are rational and real; there are also formulae that yield the required solutions. However, there is no formula for general quintic equations over the rationals in terms of radicals; this is known as the Abel–Ruffini theorem, first published in 1824, which was one of the first applications of group theory in algebra. This result also holds for equations of higher degrees.

As a practical matter, exact analytic solutions for polynomial equations are often unnecessary, and so numerical methods such as Laguerre's method or the Jenkins-Traub method are probably the best way of obtaining solutions to general quintics and higher degree polynomial equations that arise in practice. However, analytic solutions are sometimes useful for certain applications, and many mathematicians have tried to develop them.

Solvable quintics

Some fifth degree equations can be solved by factorizing into radicals, for example $x^{5}-x^{4}-x+1=0$ , which can be written as $(x^{2}+1)(x+1)(x-1)^{2}=0$ . Other quintics like $x^{5}-x+1=0$ cannot be factorized and solved in this manner. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory, and these techniques were first applied to finding a general criterion for determining whether any given quintic is solvable by John Stuart Glashan, George Paxton Young, and Carl Runge in 1885 (see Lazard's paper for a modern approach). They found that given any irreducible solvable quintic in Bring-Jerrard form,

x^{5}+ax+b=0

must have the following form:

x^{5}+{\frac {5\mu ^{4}(4\nu +3)}{\nu ^{2}+1}}x+{\frac {4\mu ^{5}(2\nu +1)(4\nu +3)}{\nu ^{2}+1}}=0

where $\mu$ and $\nu$ are rational. In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,

x^{5}+{\frac {5e^{4}(3-4c\epsilon )}{c^{2}+1}}x+{\frac {-4e^{5}(11\epsilon +2c)}{c^{2}+1}}=0

for $\epsilon =\pm 1$ . The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression

b=(4/5)\left(a+20+2{\sqrt {(20-a)(5+a)}}\right)

where

a={\frac {5(4v+3)}{v^{2}+1}}

and using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second with $\epsilon =-1$ . It is then a necessary (but not sufficient) condition that the irreducible solvable quintic

z^{5}+a\mu ^{4}z+b\mu ^{5}=0

with rational coefficients must satisfy the simple quadratic curve

y^{2}=(20-a)(5+a)

for some rational a, y.

Since by judicious use of Tschirnhaus transformations it is possible to transform any quintic into Bring-Jerrard form, both of these parameterizations give a necessary and sufficient condition for deciding whether a given quintic may be solved in radicals.

Examples of solvable quintics

A quintic is solvable using radicals if the Galois group of the quintic (which is a subgroup of the symmetric group S(5) of permutations of a five element set) is a solvable group. In this case the form of the solutions depends on the structure of this Galois group.

A simple example is given by the equation $x^{5}-5x^{4}-10x^{3}-10x^{2}-5x-1=0$ , whose Galois group is the group F(5) generated by the permutations "(1 2 3 4 5)" and "(1 2 4 3)"; the only real solution is $x=1+{\sqrt[{5}]{2}}+{\sqrt[{5}]{4}}+{\sqrt[{5}]{8}}+{\sqrt[{5}]{16}}.$

However, for other solvable Galois groups, the form of the roots can be much more complex. For example, the equation $x^{5}-5x+12$ has Galois group D(5) generated by "(1 2 3 4 5)" and "(1 4)(2 3)" and the solution requires about 600 symbols to write.

Beyond radicals

If the Galois group of a quintic is not solvable, then the Abel-Ruffini theorem tells us that to obtain the roots it is necessary to go beyond the basic arithmetic operations and the extraction of radicals. About 1835, Jerrard demonstrated that quintics can be solved by using ultraradicals (also known as Bring radicals), the real roots of $t^{5}+t-a$ for real numbers $a$ . In 1858 Charles Hermite showed that the Bring radical could be characterized in terms of the Jacobi theta functions and their associated elliptic modular functions, using an approach similar to the more familiar approach of solving cubic equations by means of trigonometric functions. At around the same time, Leopold Kronecker, using group theory developed a simpler way of deriving Hermite's result, as had Francesco Brioschi. Later, Felix Klein came up with a particularly elegant method that relates the symmetries of the icosahedron, Galois theory, and the elliptic modular functions that feature in Hermite's solution, giving an explanation for why they should appear at all, and develops his own solution in terms of generalized hypergeometric functions.

References

Charles Hermite, "Sur la résolution de l'équation du cinquème degré",Œuvres de Charles Hermite, t.2, pp. 5-21, Gauthier-Villars, 1908.
Felix Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, trans. George Gavin Morrice, Trübner & Co., 1888. ISBN 0-486-49528-0.
Leopold Kronecker, "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite", Comptes Rendus de l'Académie des Sciences," t. LXVI, 1858 (1), pp. 1150-1152.
Blair Spearman and Kenneth S. Williams, "Characterization of solvable quintics $x^{5}+ax+b$ ", American Mathematical Monthly, Vol. 101 (1994), pp. 986-992.
Ian Stewart, Galois Theory 2nd Edition, Chapman and Hall, 1989. ISBN 0-412-34550-1. Discusses Galois Theory in general including a proof of insolvability of the general quintic.
Jörg Bewersdorff, Galois theory for beginners: A historical perspective, American Mathematical Society, 2006. ISBN 0-8218-3817-2. Chapter 8 (The solution of equations of the fifth degree) gives a description of the solution of solvable quintics $x^{5}+cx+d$ .
Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of Tschirnhaus, Bring and Jerrard," ACM SIGSAM Bulletin, Vol. 37, No. 3, September 2003, pp. 90-94.
Ehrenfried Walter von Tschirnhaus, "A method for removing all intermediate terms from a given equation," ACM SIGSAM Bulletin, Vol. 37, No. 1, March 2003, pp. 1-3.
Daniel Lazard, "Solving quintics in radicals", Olav Arnfinn Laudal, Ragni Piene, The Legacy of Niels Henrik Abel, pp. 207–225, Berlin, 2004,. ISBN 3-5404-3826-2.

External links

Mathworld - Quintic Equation - more details on methods for solving Quintics.
Solving the Quintic with Mathematica - poster on Quintic solutions
Lectures on the Icosahedron - Klein's book is available online
Solving Solvable Quintics - a method for solving solvable quintics due to David S. Dummit.
Polynomial Transformations of Tschirnhaus, Bring and Jerrard - a recent update of Tschirnhaus' paper by Victor S. Adamchik & David J. Jeffrey
A method for removing all intermediate terms from a given equation - a recent English translation of Tschirnhaus' 1683 paper.
Solving quintics by radicals by Daniel Lazard - Originally in The Legacy of Niels Henrik Abel, O. Laudal and R. Piene, editors, Springer-Verlag, 2004, pp. 207-226.