Quintic Equations
Quintic equations are polynomial equations with one variable, customarily denoted by x, which is never raised to a power greater than the fifth. Symbolically, such an equation can be written as follows: ax5 + bx4 + cx3 + dxdx2 + ex + f = 0. The problem of solving polynomial equations in general, and quintic equations in particular, has been a central theme in from antiquity to the present.
Babylonian mathematicians already knew how to solve quadratic equations (with no power of x greater than 2). In the early 1500s, Italian mathematicians discovered how to solve cubic equations (with no power of x greater than 3) and quartic equations (with no power greater than 4). In each case, the solution could be found with nothing more than the elementary operations of addition, subtraction, multiplication and division, plus the special operations of taking square roots or cube roots. This discovery suggested that a general method for solving all polynomials might be just around the corner.
However, more than 250 years went by with very little progress. Finally, in 1826, the Norwegian mathematician Niels Henrik Abel cleared up the mystery, in a novel and unexpected way: He showed that no formula had been found because no such formula could ever exist. That is, there is no "black box" that could accept the coefficients a, b, c, d, e, f as input, churn through a calculation that involves only elementary operations or n-th roots (also called "radicals"), and be guaranteed to produce, as output, a valid solution x to the above equation.
This was one of the first great impossibility theorems of mathematics, and its proof required a revolutionary approach. To prove that there is no "black box" that can solve a polynomial, one must study black boxes. That is, it is necessary to study the operations themselves: addition, subtraction, multiplication, division, and n-th roots.
The first four of these operations define a mathematical structure known as a field; and the operation of finding an n-th root amounts to constructing what is called a "cyclic extension" of a field. Quadratic, cubic, and quartic equations are solvable by radicals because their solutions always lie in cyclic extensions of cyclic extensions of cyclic extensions of the field of rational numbers. But for quintic equations, that is no longer true.
In 1830, Evariste Galois developed the method that mathematicians still use to study extension fields. Any extension field, Galois proved, possesses symmetries. For example, in a quadratic extension the two square roots can be interchanged without altering the underlying field. (This is the source of the ambiguous ± sign in the quadratic formula.) These symmetries are described by a mathematical structure called a Galois group, and the structure of the extension field is exactly mirrored in the structure of its Galois group. If the field is constructed by cyclic extensions, then the Galois group can be "deconstructed" by cyclic factor groups. But Galois realized that some Galois groups cannot be decomposed at all; they spring into existence fully formed, as it were. These are called simple groups, and the smallest one, called A5, has 60 elements. It is too large to be the Galois group for a quartic polynomial (which can have 24 elements at most), but it can be the Galois group for a quintic. In fact, later mathematicians have shown that any quintic polynomial p(x) with rational coefficients, which cannot be split into smaller polynomials, and which has exactly three real-number solutions, must have A5 as its Galois group. It follows that the equation p(x) = 0 cannot be solved by radicals.
But quintic equations can be solved in other ways. One way is simply to add more tools to the "black box." In 1858, Charles Hermite showed how to solve all quintic polynomials by using theta functions, and Felix Klein in 1888 did the same with hypergeometric functions. But for working mathematicians and engineers, who may not want to learn about specialized and unfamiliar classes of functions, a more practical approach is to use approximate numerical methods. Equation-solving routines are now a standard part of computer algebra packages. Research is continuing on robust and efficient computer-based methods for solving not only quintic polynomials, but polynomials involving much higher powers of x.
Why, when adequate methods exist for solving even more complicated polynomials, do mathematicians still attach so much importance to theorems about inadequate methods? One might as well ask why baseball pitchers continue to throw the ball by hand, when a howitzer would work much better. Mathematics, like the game of baseball, has a certain integrity to it that can be spoiled if the "rules of the game" are changed too drastically. Moreover, the algebraic approach teaches us about the structure of all polynomials, while the brute-force computer approach only teaches us about one polynomial at a time.
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