Solving Quintic Equations

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Solving Quintic Equations

Overview

By the nineteenth century, mathematicians had long been interested in solving equations called polynomials. However, Paolo Ruffini (1765-1822) and Niels Abel (1802-1829) proved that some polynomials could not be solved by previously known methods. Partly in response, Evariste Galois (1811-1832) developed a new way of analyzing and working with these types of equations. This method is called group theory, and it was to have implications in other scientific fields, such as mineralogy, physics, and chemistry.

Background

Polynomial equations are used in almost every branch of mathematics and science. An example of a polynomial equation is 3x² + 4x + 5 = 0. This equation is called a second degree polynomial because the highest power of x it contains is 2. The degree of a polynomial indicates the number of solutions it has. A number is said to be a solution of a polynomial equation if substituting it into the equation makes the equation true. For instance, the number 7 is a solution of the equation x + 5 = 12. By the nineteenth century, mathematicians had already discovered ways to solve second, third, and fourth degree polynomial equations. They next turned their attention to solving fifth degree, or quintic, equations. (An example of a quintic equation is 6x⁵ + 3x⁴ + 3x² + 5x + 6 = 0.)

The fundamental theorem of algebra would come to be important in finding solutions to quintic equations. Carl Gauss (1777-1855), who is sometimes referred to as the founder of modern mathematics, proved this theorem in 1801. Gauss's theorem deals with the relationship between the coefficients of a polynomial equation and its solutions. (For the polynomial 4x² + 7x = 0, the coefficients are the numbers 4 and 7.) Specifically, the fundamental theorem of algebra concerns polynomials with complex coefficients. Complex numbers consist of two parts: a real part and an imaginary part. The real numbers are the positive numbers, negative numbers, and zero. Imaginary numbers are the product of a real number and i. (i represents the square root of -1.) The number 5 + 3i is an example of a complex number; 5 is the real part, and 3i is the imaginary part. The number 7 is also a complex number; 7 is equal to 7 + 0i. Gauss's fundamental theorem of algebra states that every polynomial equation with complex coefficients has at least one complex solution.

Polynomial equations of degree less than five are said to be solvable by radicals. This means that they can be solved by a combination of addition, subtraction, multiplication, division, and the taking of roots. In 1796 Paolo Ruffini, an Italian mathematician, proposed that quintic equations could not be solved by radicals. However, because of the fundamental theorem of algebra, Ruffini knew that solutions to quintics must exist. Therefore, his hypothesis suggested that much more complicated mathematics would be necessary to solve these equations. Ruffini attempted a proof of his proposal in 1799, but he was not entirely successful. In addition, much of his work was so complicated that even leading mathematicians could not understand it. Ruffini published additional proofs in attempts to convince others of his discovery. Yet, he continued to have great difficulty in achieving recognition for his work.

In 1824 Norwegian mathematician Niels Abel also presented a proof that, in general, quintic equations could not be solved by radicals. Abel sent a copy of his paper to Carl Gauss, who, like the mathematicians who had seen Ruffini's proofs, failed to realize its importance. Within five years, however, Abel's work became known and accepted throughout the European mathematical community. Abel also showed that in some special cases, quintic equations could be solved by radicals. These equations are now called Abelian equations. For instance, it is easy to see that the quintic equation x⁵ - 1 = 0 is true when x = 1. This fact made Abel wonder whether if there is a way to determine whether a quintic equation is easily solvable. However, he died at the age of 26 before he could begin to investigate this question.

The French mathematician Evariste Galois took up Abel's work. In particular, Galois studied Abel's ideas concerning groups. A group is a set of numbers that can be combined in pairs so that the resulting numbers are also in the set. (For example, integers form a group for the operation of addition. Whenever two integers are added, the result is always another integer.) Galois developed what became known as the group theory of algebra. Group theory is a branch of mathematics concerned with identifying groups and studying their properties. His work on group theory was not widely recognized until it was published in 1846, 14 years after his death.

Impact

Group theory led to an entirely different way of searching for and analyzing the solutions of polynomial equations. It involved examining the permutations of the solutions of an equation. A permutation is a combination of a group of objects in which the order of the objects is important. (For example, the permutations of the letters A and B are AB and BA.) All of the permutations of the solutions of an equation form a group. These permutations can then be combined in different ways to form subgroups. By analyzing the ways in which these subgroups are related, Gauss could determine whether or not a polynomial equation was solvable by radicals.

Galois's group theory is one indication of a major transition that occurred in the field of mathematics during the nineteenth century. Rather than performing calculations to solve a particular problem (such as finding the two solutions of the equation x² + 5x - 6 = 0), mathematicians began to work with extremely complicated analyses of general problems (such as determining which types of polynomials are solvable by radicals). Galois realized that his method of using groups was extremely theoretical, and he did not intend for it to be a practical method of solving equations. In fact, Galois's analyses of the permutations of the solutions of equations was performed without actually knowing the numerical values of the solutions themselves.

In 1870 French mathematician Camille Jordan (1838-1922) published an edited version of Galois's theory in his book Treatise on Substitutions and Algebraic Equations. Many modern concepts of group theory first appeared in Jordan's work. For instance, he defined a solvable group as a group that belonged to an equation solvable by radicals. Jordan also answered the question posed by Abel, that of determining which quintic equations are solvable by radicals. Jordan concluded that a quintic equation is solvable by radicals if its solutions form a solvable group.

Charles Hermite (1822-1901), a French mathematician, published a solution to quintic equations in 1858. His solution involved the use of elliptic functions. A function defines a relationship between an independent variable and a dependent variable. For example, y = 3x + 5 is a function in which x is the independent variable and y is the dependent variable. An elliptic function can be used to calculate the perimeter of an ellipse. The dependent variable of an elliptic function is complex (in other words, it has a real part and an imaginary part). Therefore, Hermite showed that complex functions could be used to solve quintic equations.

In 1878 Ludwig Kiepert, a German mathematician, wrote an article describing a systematic procedure that could be used to solve quintic equations based upon Galois's group theory and Hermite's use of elliptic functions. However, Kiepert's procedure, as well as those of other mathematicians, were largely impractical because the mathematics involved were simply too complex to be performed at that time. Not until the late twentieth century and the development of computers could such procedures be put to practical use.

Another application of Galois's group theory is that it can be used to study the symmetry of physical objects. An object has symmetry if a change in its position in space seems to leave it unmoved. For example, when a square is rotated 180°, it seems not to have moved at all. A rotation of 180° is therefore an element of symmetry for a square. Elements of symmetry may include rotation, reflection, and translation.

Felix Klein (1849-1925) was one of the first mathematicians to use group theory and the solutions of polynomial equations to study the symmetry of physical objects. Specifically, he studied quintic equations. A quintic polynomial has five solutions. The group composed of the permutations of these five solutions consists of 120 elements. (There are 5! or 120 permutations of five objects taken five at a time.) An icosahedron is a three-dimensional figure with 20 faces and 120 elements of symmetry. These elements of symmetry form a group. Therefore, the group of a quintic equation can be used to study the symmetry group of an icosahedron, and vice versa. Klein first described this relationship in 1884.

Quintic equations are not the only polynomials that can be used to study symmetry. For example, quartic, or fourth degree, polynomials can be used to analyze the symmetry of tetrahedrons. A tetrahedron has four faces, each of which is an equilateral triangle. The permutation group of quartic polynomials has 4! or 24 elements, and the symmetry group of a tetrahedron also has 24 elements.

Auguste Bravais (1811-1863), a French physicist and mineralogist, used group theory as it related to symmetry to determine the structure of crystals. The atoms in a crystal are arranged in a definite pattern. These arrangements exhibit elements of symmetry. Bravais analyzed the permutations of the solutions of polynomials in order to study this symmetry. In 1849 he published a paper in which he proposed the 32 classes of molecular structures found in crystals.

In the twentieth century, physicists used group theory to study the interactions of sub-atomic particles. By analyzing symmetry, they could determine which interactions between particles were possible and which were not. Chemists also studied symmetry to determine the arrangement of atoms in molecules. For example, they determined that a molecule of methane has the shape of a tetrahedron. A methane molecule consists of one atom of carbon and four atoms of hydrogen. The carbon atom is located at the center, and the hydrogen atoms are positioned around it at the four corners of a tetrahedron. Therefore, the symmetry of a molecule of methane can be described by a quartic polynomial equation. There are also molecules that can be studied with quintic equations. For example, a chemical called o-carborane consists of two atoms of carbon, 10 atoms of boron, and 12 atoms of hydrogen. The two carbon atoms are at the center of the molecule, and the boron and hydrogen atoms form the corners of an icosahedron. Today, scientists and mathematicians in many fields continue to find new applications of group theory.

STACEY R. MURRAY