ABSTRACT
In the coming chapters, we are going to use the group theory discussed so far to see how to solve polynomial equations. What “solving” an equation means is a rather delicate question. People have known how to write solutions for a quadratic equation in terms of the square root of its discriminant for some 4000 years, and today everyone learns the formula in high school. It is simple and very useful. In the Renaissance, similar formulas were discovered for cubics and quartics. However, they are much more complicated and much less useful. Early in the 19th century, it was realized that for equations of degree greater than 4, formulas do not even exist for solutions in terms of radicals. At the same time, several mathematicians noticed that the symmetries of an equation, as we discussed in some examples in Chapter 7, tell you many interesting and profound things about its solutions. This point of view has been developed with great success in the past two centuries and will be the theme of the remainder of this book. If you are interested in the history of these ideas, the first part of van der Waerden’s History of Algebra [9] is a good reference.
