Polynomial Equations

Preview

The first phase in the history of algebra was the search for solutions of polynomial equations. The “degree of difficulty” of an equation corresponds rather neatly to the degree of the corresponding polynomial. Linear equations are easily solved, and 2000 years ago the Chinese were even able to solve n equations in n unknowns by the method we now call “Gaussian elimination.” Quadratic equations are harder to solve, because they generally require the square root operation. But the solution—essentially the same as that taught in high schools today—was discovered independently in many cultures more than 1000 years ago. The first really hard case is the cubic equation, whose solution requires both square roots and cube roots. Its discovery by Italian mathematicians in the early 16th century was a decisive breakthrough, and equations quickly became the language of virtually all mathematics. (See, for example, analytic geometry in Chapter 7 and calculus in Chapter 9.) Despite this breakthrough, the problem of polynomial equations remained incompletely solved. The obstacle was the quintic equation—the general equation of degree 5. In the 1820s it finally became clear that the quintic equation is not solvable in the sense that equations of lower degree are solvable. But explaining why this is so requires a new, and more abstract, concept of algebra (see Chapter 19).