Quintic equation

# Quintic equation

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a quintic function is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of the form

where a, b, c, d, e and f are members of a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, typically the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s, the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s or the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, and a is nonzero. In other words, a quintic function is defined by a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

of degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...

five.

Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

of the form:

If a is zero but one of the other coefficients is non-zero, the equation is classified as either a quartic equation, cubic equation, quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

or linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

.

Because they have an odd degree, normal quintic functions appear similar to normal cubic function
Cubic function
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...

s when graphed, except they may possess an additional local maximum
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...

and local minimum each. The derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

of a quintic function is a quartic function
Quartic function
In mathematics, a quartic function, or equation of the fourth degree, is a function of the formf=ax^4+bx^3+cx^2+dx+e \,where a is nonzero; or in other words, a polynomial of degree four...

.

## 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
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

, cubic and quartic equations by factorization
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

s is fairly straightforward, no matter whether the roots are rational or irrational, real or complex; 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
Abel–Ruffini theorem
In algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...

, first published in 1824, which was one of the first applications of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

in algebra. This result also holds for equations of higher degrees. An example quintic whose roots cannot be expressed by radicals is This quintic is in Bring–Jerrard normal form.

As a practical matter, exact analytic solutions for polynomial equations are often unnecessary, and so numerical methods such as Laguerre's method
Laguerre's method
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 equation\ p = 0 for a given polynomial p...

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, , which can be written as , or , which has as solution. Other quintics like cannot be solved by radicals. Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...

developed techniques for determining whether a given equation could be solved by radicals which gave rise to group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

and Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

. Applying these techniques, Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

has found a general criterion for determining whether any given quintic is solvable. This criterion is the following.

Given the equation,
the Tschirnhaus transformation
Tschirnhaus transformation
In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. It may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a...

, which depresses the quintic, gives the equation
,

where

Both equations are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial , named Cayley resolvent, has a rational root in z, where

and

In 1888, George Paxton Young described how to solve a solvable quintic equation, without providing an explicit formula; Lazard (2004) writes out the three-page formula.

During the second half of 19th century, John Stuart Glashan, George Paxton Young, and Carl Runge found that any irreducible
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

quintic with rational coefficients in Bring
Erland Samuel Bring
Erland Samuel Bring was a Swedish mathematician.Bring studied at Lund University between 1750 and 1757. In 1762 he obtained a position of a reader in history and was promoted to professor in 1779...

-Jerrard
George Jerrard
George Birch Jerrard was a British mathematician.He studied at Trinity College, Dublin from 1821 to 1827. His main work was on the theory of equations, where he was reluctant to accept the validity of the work of Niels Henrik Abel on the insolubility of the quintic equation by radicals...

form,

is solvable by radicals if and only if either a = 0 or it is of the following form:

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

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. It is then a necessary (but not sufficient) condition that the irreducible solvable quintic

with rational coefficients must satisfy the simple quadratic curve

for some rational .

The substitution , in Spearman-Williams parameterization allows to not exclude the special case a = 0, giving the following result:

If a and b are rational numbers, the equation is solvable by radicals if either its left hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers l and m such that.

### Examples of solvable quintics

A quintic is solvable using radicals if the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

of the quintic (which is a subgroup of the symmetric group S(5) of permutations of a five element set) is a solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

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

A simple example is given by the equation , 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 has Galois group D(5) generated by "(1 2 3 4 5)" and "(1 4)(2 3)" and the solution requires more symbols to write. Define,

where φ is the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

, then the only real solution is exactly given by,

Or equivalently,

where the yi are the four roots of the quartic equation,

In general, if an equation P(x) = 0 of prime degree p with rational coefficients is solvable in radicals, there is an auxiliary equation Q(y) = 0 of degree (p-1) also with rational coefficients that can be used to solve the former.

However, it is possible some of the roots of Q(y) = 0 are rational (like in the F(5) example given above) or some as zero, like the solvable DeMoivre quintic,

where the auxiliary equation has two zero roots and is essentially just the quadratic
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

,

such that the five roots of the DeMoivre quintic are given by,

where ωk is any of the five 5th roots of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

. This can be easily generalized to construct a solvable septic
Septic equation
In mathematics, a septic equation, heptic equation or septimic equation is a polynomial equation of degree seven. It is of the form:ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,...

and other odd degrees, not necessarily prime.

Here is a list of known solvable quintics:

There are infinitely many solvable quintics in Bring-Jerrard form which have been parameterized in preceding section.

Up to the scaling of the variable, there is exactly five solvable quintics of the shape , which are (where s is a scaling factor):

Paxton Young (1888) gave a number of examples, some of them being reducible, having a rational root: >
 Solution: Reducible: −8 is a root Reducible: −4 is a root Reducible : -8 is a root Reducible : 8 is a root

An infinite sequence of solvable quintics may be constructed, whose roots are sums of n-th roots of unity, with n = 10 k + 1 being a prime number:
 Roots: Root: Root: Root: Root:

There are also two parameterized families of solvable quintics:
The Kondo–Brumer quintic,

and the family depending on the parameters

where

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
George Jerrard
George Birch Jerrard was a British mathematician.He studied at Trinity College, Dublin from 1821 to 1827. His main work was on the theory of equations, where he was reluctant to accept the validity of the work of Niels Henrik Abel on the insolubility of the quintic equation by radicals...

demonstrated that quintics can be solved by using ultraradicals (also known as Bring radical
In algebra, a Bring radical or ultraradical of a complex number a is a root of the polynomialx^5+x+a. \,In algebra, a Bring radical or ultraradical of a complex number a is a root of the polynomialx^5+x+a. \,In algebra, a Bring radical or ultraradical of a complex number a is a root...

s), the real roots of for real numbers . In 1858 Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

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 function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s. At around the same time, Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...

, using group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

developed a simpler way of deriving Hermite's result, as had Francesco Brioschi
Francesco Brioschi
Francesco Brioschi was an Italian mathematician.Brioschi was born in Milan in 1824. From 1850 he taught analytical mechanics in the University of Pavia. After the Italian unification in 1861, he was elected depute in the Parliament of Italy and then appointed twice secretary of the Education...

. Later, Felix Klein
Felix Klein
Christian Felix Klein 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...

came up with a particularly elegant method that relates the symmetries of the icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group 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. Similar phenomena occur in degree 7 (septic equation
Septic equation
In mathematics, a septic equation, heptic equation or septimic equation is a polynomial equation of degree seven. It is of the form:ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,...

s) and 11, as studied by Klein and discussed in icosahedral symmetry: related geometries.