Preface

Even to this day, I recall my first algebra course as wonderfully delightful and

exciting. In my youthful pride, I also recall secretly fearing that when I told people I

was studying algebra, they would think I was doing the type of mathematics taught

to twelve-year-olds. After all, once you can solve 2x = 4 and know how to use the

quadratic equation, what else is there?

In fact, algebra is a large subfield of modern mathematics that has been in-

tensely researched over the last few hundred years. It is very different from the

type of mathematics that most students see during the bulk of their college years.

Usually the majority of undergraduate classes are devoted to studying the subfield

of mathematics known as analysis, which deals with limits, derivatives, integrals,

and all sorts of related concepts.

In contrast, algebra is the study of mathematical systems that, in some sense,

generalize the behavior of numbers. For example, the set of integers, the set of

polynomials, and the set of n × n matrices are all very different types of objects.

Nevertheless, it makes sense to “add” and “multiply” in each of these cases. More-

over in each of these settings, the formal behavior of addition and multiplication is

strikingly similar. As a result, it is possible to abstract the properties of addition

and multiplication in order to prove a single theorem applying simultaneously to

each different case. This allows for a great deal of eﬃciency.

More importantly, these “algebraic” techniques frequently make it possible to

tackle problems that are otherwise completely inaccessible. As a remarkable ex-

ample, the ancient Greeks were interested in constructions using only a compass

and straightedge. For instance, they could construct a regular 3-gon, 4-gon, 5-gon,

6-gon, 8-gon, and 10-gon, but they never succeeded in constructing a regular 7-gon,

9-gon, or 11-gon. For centuries afterward, geometers tried and failed to repair this

gap. In fact, it took a few thousand years and the advent of algebraic techniques

for people to figure out what was going on. In 1837, Pierre Wantzel succeeded in

showing that the diﬃculty did not lie with a failure of human ingenuity, but rather

that it was impossible to construct a regular 7-gon, 9-gon, or 11-gon (Exercise

4.179).

This book is intended as a first course in algebra. For the most part, the only

formal prerequisite is the ability to add and multiply integers. In principal, my

nine-year-old daughter is therefore more than prepared to study algebra and ought

to have no problem jumping right in on page one. Excepting the rare genius, of

course, this is a big fallacy. Generally speaking, algebra is considered one of the

most diﬃcult courses taken by an undergraduate mathematics major. While it is

technically true that the greatest prerequisite for this material is familiarity with

the integers, the true prerequisite is mathematical maturity. For many students, a

“how to prove things” course is a good way to obtain this experience, though this

xi