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

In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. It is of the form: Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 File links The following pages link to this file: Polynomial Quintic function Categories: GFDL images ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 File links The following pages link to this file: Polynomial Quintic function Categories: GFDL images ... In mathematics, a critical point (or critical number) is a point on the domain of a function where: one dimension: the derivative is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

where a,b,c,d,e,f are members of a field, (typically the rational numbers, the real numbers or the complex numbers), and . In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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. Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ... A graph illustrating local min/max and global min/max points In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the... For a non-technical overview of the subject, see Calculus. ... Polynomial of degree 4: f(x) = (x+4)(x+1)(x-1)(x-3)/14+0. ...

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. A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... ... In mathematics, an nth root of a number a is a number b, such that bn=a. ... The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ... 1824 was a leap year starting on Thursday (see link for calendar). ... Group theory is that branch of mathematics concerned with the study of groups. ...

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. In numerical analysis, Laguerres method is a root-finding algorithm tailored to polynomials. ... The Jenkins-Traub method is a complicated root-finding algorithm for real polynomials which is widely considered to be reliable, is used in a number of numerical analysis packages and has Fortran and C implementations in the public domain. ...

Solvable quintics

Some fifth degree equations can be solved by factorizing into radicals, for example x⁵ − x⁴ − x + 1 = 0, which can be written as (x² + 1)(x + 1)(x − 1)² = 0. Other quintics like x⁵ − 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, Galois at the age of fifteen from the pencil of a classmate. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... Carle David Tolmé Runge (August 30, 1856 – January 3, 1927) was a German mathematician, physicist, and spectroscopist. ... 1885 (MDCCCLXXXV) is a common year starting on Thursday of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ...

x⁵ + ax + b = 0

must have the following form:

where μ and ν are rational. In 1994, Blair Spearman and Kenneth S. Williams gave an alternative, Year 1994 (MCMXCIV) was a common year starting on Saturday (link will display full 1994 Gregorian calendar). ...

for . The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression

where

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

z⁵ + aμ⁴z + bμ⁵ = 0

with rational coefficients must satisfy the simple quadratic curve

y² = (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. In mathematics, a Tschirnhaus transformation is a type of mapping on polynomials. ...

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. In mathematics, a Galois group is a group associated with a certain type of field extension. ... In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ...

A simple example is given by the equation x⁵ − 5x⁴ − 10x³ − 10x² − 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

However, for other solvable Galois groups, the form of the roots can be much more complex. For example, the equation x⁵ − 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

Main article: Bring radical

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 t⁵ + 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. In algebra, a Bring radical or ultraradical is a real zero of the polynomial Where a is a complex number. ... George Birch Jerrard (1804 – 1863) was a British mathematician. ... In algebra, a Bring radical or ultraradical is a real zero of the polynomial Where a is a complex number. ... In algebra, a Bring radical or ultraradical is a real zero of the polynomial Where a is a complex number. ... Charles Hermite (pronounced in IPA, ) (December 24, 1822 – January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... In mathematics, theta functions are special functions of several complex variables. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Group theory is that branch of mathematics concerned with the study of groups. ... Francesco Brioschi. ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...

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 x⁵ + 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 x⁵ + 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.

See also

• Solvable group
• Theory of equations

In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ... In mathematics, the theory of equations comprises a major part of traditional algebra. ...

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.