Chapter 4

Polynomials

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

—Heinrich Hertz

The study of polynomials is the oldest branch of algebra. The Hindus knew how to solve quadratics in 600 BC, and the Babylonians by then had developed considerable skill at algebraic manipulation and were using special cases of the quadratic formula. However, symbolic algebra in the form we know it today developed in Arabia between 600 and 1000 AD. They were solving cubic equations and, in the work of al-Khowarizmi (c.825), were starting to identify geometric magnitudes with numbers. These efforts led them to the familiar formulas for areas, volumes, and the like. By Descartes' time (1596–1650), analytic geometry was well understood, so that the computational power of algebra and the intuitive power of geometry could each enhance the other.

Subsequently, the theory of equations attracted the best mathematicians. Euler and Lagrange considered the problem of finding a general formula, analogous to the quadratic formula, for the roots of any quintic polynomial (degree 5). Their work led to the epoch making discovery of Abel who, in 1823, showed that no such formula exists. Later Galois showed it is impossible for any polynomial of degree 5 or more, and brought groups into the picture.

The ...