# Beyond the Quartic Equation

Quintic Equation

This procedure produces extraneous solutions, but when we have found the correct ones by numerical means we can also write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable — an algebraic solution of the general quintic. Many other characterizations of the Bring radical have been developed, the first of which is in terms of elliptic modular functions by Charles Hermite in , and further methods later developed by other mathematicians.

However, certain classes of quintic equations can be solved in this manner. Get A Copy. Its discriminant is. Chapter 4 present elliptic functions as a generalization of radicals and covers theta functions, which can be used to compute values of elliptic functions. The answer is Yes , an equation solvable in radicals always has a trigonometric solution. Be the first to ask a question about Beyond the Quadratic Formula. Student 13 , 84,

In , Charles Hermite [8] published the first known solution to the general quintic equation in terms of elliptic transcendents, and at around the same time Francesco Brioschi [9] and Leopold Kronecker [10] came upon equivalent solutions. Hermite arrived at this solution by generalizing the well-known solution to the cubic equation in terms of trigonometric functions and finds the solution to a quintic in Bring — Jerrard form:. He observed that elliptic functions had an analogous role to play in the solution of the Bring — Jerrard quintic as the trigonometric functions had for the cubic. If and are the periods of an elliptic integral of the first kind:.

If n is a prime number , we can define two values u and v as follows:. The modular equation of the sixth degree may be related to the Bring — Jerrard quintic by the following function of the six roots of the modular equation:. The five quantities , , , , are the roots of a quintic equation with coefficients rational in :.

To do this, let. Any of these roots may be used as the elliptic modulus for the purposes of the method. The value of may be easily obtained from the elliptic modulus by the relations given above. The roots of the Bring — Jerrard quintic are then given by:.

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It may be seen that this process uses a generalization of the nth root , which may be expressed as:. The Hermite — Kronecker — Brioschi method essentially replaces the exponential by an elliptic modular function, and the integral by an elliptic integral. Kronecker thought that this generalization was a special case of a still more general theorem, which would be applicable to equations of arbitrarily high degree. This derivation due to M. Glasser [13] generalizes the series method presented earlier in this article to find a solution to any trinomial equation of the form:.

In particular, the quintic equation can be reduced to this form by the use of Tschirnhaus transformations as shown above. Let , the general form becomes:. A formula due to Lagrange states that for any analytic function , in the neighborhood of a root of the transformed general equation in terms of , above may be expressed as an infinite series :. If we let in this formula, we can come up with the root:.

Quartic Equations Part 2

By the use of the Gauss multiplication theorem the infinite series above may be broken up into a finite series of hypergeometric functions :. Applying this method to the reduced Bring — Jerrard quintic, define the following functions:.

The roots of the quintic are thus:. James Cockle [14] and Robert Harley [15] developed, in , a method for solving the quintic by means of differential equations. They consider the roots as being functions of the coefficients, and calculate a differential resolvent based on these equations. The Bring — Jerrard quintic is expressed as a function:. The function must also satisfy the following four differential equations:. The solution of the differential resolvent, being a fourth order ordinary differential equation, depends on four constants of integration , which should be chosen so as to satisfy the original quintic.

This is a Fuchsian ordinary differential equation of hypergeometric type, [16] whose solution turns out to be identical to the series of hypergeometric functions that arose in Glasser's derivation above. This method may also be generalized to equations of arbitrarily high degree, with differential resolvents which are partial differential equations , whose solutions involve hypergeometric functions of several variables. In , Peter Doyle and Curt McMullen derived an iteration method [21] that solves a quintic in Brioschi normal form:.

Iterate on a random starting guess until it converges. Call the limit point and let.

### Samenvatting

The objective of this book is to present for the first time the complete algorithm for roots of the general quintic equation with enough background information to. Beyond the Quartic Equation (Modern Birkhäuser Classics) 1st ed. The book includes an initial introductory chapter on group theory and symmetry, Galois theory and Tschirnhausen transformations, and some elementary properties of elliptic function in order to make some of the.

The two polynomial functions and are as follows:. This iteration method produces two roots of the quintic. The remaining three roots can be obtained by using synthetic division to divide the two roots out, producing a cubic equation. It is to be noted that due to the way the iteration is formulated, this method seems to always find two complex conjugate roots of the quintic even when all the quintic coefficients are real and the starting guess is real.

This iteration method is derived by from the symmetries of the icosahedron and is closely related to the method Felix Klein describes in his book. In this article, the Bring radical of a is denoted Normal forms The quintic equation is rather difficult to obtain solutions for directly, with five independent coefficients in its most general form: The various methods for solving the quintic that have been developed generally attempt to simplify the quintic using Tschirnhaus transformations to reduce the number of independent coefficients.

In Bring found a way around this by using a quartic Tschirnhaus transformation to relate the roots of a principal quintic to those of a Bring — Jerrard quintic: The extra parameter this fourth-order transformation provides allowed Bring to decrease the degrees of the other parameters. A similar transformation suffices to reduce the equation to which is the form required by the Hermite-Kronecker-Brioschi method, Glasser's method, and the Cockle-Harley method of differential resolvents described below. Brioschi normal form There is another one-parameter normal form for the quintic equation, known as Brioschi normal form : which can be derived by using the following rational Tschirnhaus transformation to relate the roots of a principal quintic to a Brioschi quintic.

Series representation A Taylor series for Bring radicals, as well as a representation in terms of hypergeometric functions can be derived as follows. The series for can then be obtained by reversion of the Taylor series for which is simply , giving: where the absolute values of the coefficients are sequence A in the OEIS. This gives The series converges for and can be analytically continued in the complex plane. The above result can be written in hypergeometric form as: [5] Compare with the hypergeometric functions that arise in Glasser's derivation and the method of differential resolvents below.

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Solution of the general quintic We now may express the roots of any polynomial in terms of the Bring radical as and its four conjugates. Other characterizations Many other characterizations of the Bring radical have been developed, the first of which is in terms of elliptic modular functions by Charles Hermite in , and further methods later developed by other mathematicians.

The Hermite—Kronecker—Brioschi characterization In , Charles Hermite [8] published the first known solution to the general quintic equation in terms of elliptic transcendents, and at around the same time Francesco Brioschi [9] and Leopold Kronecker [10] came upon equivalent solutions. Hermite arrived at this solution by generalizing the well-known solution to the cubic equation in terms of trigonometric functions and finds the solution to a quintic in Bring — Jerrard form: into which any quintic equation may be reduced by means of Tschirnhaus transformations as has been shown.

If and are the periods of an elliptic integral of the first kind: the elliptic nome is given by: and With define the two elliptic modular functions : where and similar are Jacobi theta functions. To do this, let and calculate the required elliptic modulus by solving the quartic equation: The roots of this equation are: where [11] note that some important references erroneously give it as [7] [8].

The roots of the Bring — Jerrard quintic are then given by: for. It may be seen that this process uses a generalization of the nth root , which may be expressed as: or more to the point, as The Hermite — Kronecker — Brioschi method essentially replaces the exponential by an elliptic modular function, and the integral by an elliptic integral. Glasser [13] generalizes the series method presented earlier in this article to find a solution to any trinomial equation of the form: In particular, the quintic equation can be reduced to this form by the use of Tschirnhaus transformations as shown above.

Applying this method to the reduced Bring — Jerrard quintic, define the following functions: which are the hypergeometric functions that appear in the series formula above. The roots of the quintic are thus: This is essentially the same result as that obtained by the following method. The method of differential resolvents James Cockle [14] and Robert Harley [15] developed, in , a method for solving the quintic by means of differential equations.

The Bring — Jerrard quintic is expressed as a function: and a function is to be determined such that: The function must also satisfy the following four differential equations: Expanding these and combining them together yields the differential resolvent: The solution of the differential resolvent, being a fourth order ordinary differential equation, depends on four constants of integration , which should be chosen so as to satisfy the original quintic.

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