Graph of a polynomial of degree 5, with 4 critical points. In Mathematics, a critical point (or critical number) is a point on the domain of a function where one dimension

In mathematics, a quintic equation is a polynomial equation of degree five. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent When a Polynomial is expressed as a sum or difference of terms (e 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 $a \neq 0$. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

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. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that The derivative of a quintic function is a quartic function. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change A quartic function is a function of the form f(x=ax^4+bx^3+cx^2+dx+e \ with nonzero a; or in other words a Polynomial

## 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. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, factorization ( also factorisation in British English) or factoring is the decomposition of an object (for In Mathematics, an n th root of a Number a is a number b such that bn = a. 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. The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to Polynomial equations of Year 1824 ( MDCCCXXIV) was a Leap year starting on Thursday (link will display the full calendar of the Gregorian Calendar (or a Leap year Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. 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. In Numerical analysis, Laguerre's method is a Root-finding algorithm tailored to Polynomials In other words Laguerre's method can be used to solve numerically The Jenkins-Traub algorithm for polynomial zeros is a fast globally convergent iterative method 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 x5x4x + 1 = 0, which can be written as (x2 + 1)(x + 1)(x − 1)2 = 0. Other quintics like x5x + 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). In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory Carl David Tolmé Runge (pronounced /ˈʀuŋˌge/ ( August 30 1856 &ndash January 3 1927) was a German Mathematician, Year 1885 ( MDCCCLXXXV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Common They found that given any irreducible solvable quintic in Bring-Jerrard form,

x5 + 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 μ and ν are rational. 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 set In 1994, Blair Spearman and Kenneth S. Year 1994 ( MCMXCIV) was a Common year starting on Saturday (link will display full 1994 Gregorian calendar) Williams gave an alternative,

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

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

$b \equiv {4 \over 5} \left(a+20+2\sqrt{(20-a)(5+a)}\right)$

where

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

and using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second with ± functioning as −. 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, developed by Ehrenfried Walther von Tschirnhaus in 1683 is a type of mapping on Polynomials It may be defined

### 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 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 In this case the form of the solutions depends on the structure of this Galois group.

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 $x=1+\sqrt[5]{2}+\sqrt[5]{4}+\sqrt[5]{8}+\sqrt[5]{16}.$

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

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. In Algebra, a Bring radical or ultraradical is a root of the polynomial x^5+x+a \ where a is a Complex number The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to Polynomial equations of About 1835, Jerrard demonstrated that quintics can be solved by using ultraradicals (also known as Bring radicals), the real roots of t5 + ta for real numbers a. George Birch Jerrard (1804 &ndash 1863 was a British Mathematician. In Algebra, a Bring radical or ultraradical is a root of the polynomial x^5+x+a \ where a is a Complex number In Algebra, a Bring radical or ultraradical is a root of the polynomial x^5+x+a \ where a is a Complex number 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. Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did In Mathematics, theta functions are Special functions of Several complex variables. In Mathematics, Klein's j -invariant, regarded as a function of a complex variable &tau is a Modular function defined on the This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. At around the same time, Leopold Kronecker, using group theory developed a simpler way of deriving Hermite's result, as had Francesco Brioschi. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Francesco Brioschi ( December 22 1824 – December 13 1897) was an Italian Mathematician. 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. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory

## Linear algebraic methods

The quintic equation can be solved by creating a companion matrix of the quintic equation and calculating the eigenvalues of said matrix. In Linear algebra, the companion matrix of the Monic polynomial p(t=c_0 + c_1 t + \dots + c_{n-1}t^{n-1} + t^n is the Square In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

## 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. XLVI, 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.
• 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.