Mathematical research ﬂourished in the twentieth century both in

quality and quantity, and shows no signs of abating in the twenty-

ﬁrst century. Yet many people still have the misconceptions that

•

everything important in mathematics has already been

discovered, and

•

mathematics is of interest only for its applications to

science and technology.

Nevertheless, the general audience for current ideas in pure

mathematics is clearly growing, as evidenced by a spate of recent

books. Based on the level of some of these books, it seems clear that

a segment of this reading public also desires to go more deeply into

the mathematics than was typical a generation ago.

This book is a popular exposition of cutting-edge research in

one important area of mathematics, number theory. In it, we

hope both to share the excitement and to help increase popular

awareness of the intrinsic beauty of contemporary explorations in

pure mathematics.

We have in mind a broad audience, centered principally on

those who have studied calculus. Though calculus is not used in

this book, the amount of mathematical maturity needed to follow

everything in our book probably requires that level of mathematical

experience. Professional mathematicians who are not expert in

number theory but who want to learn something of its latest

methods should also ﬁnd something worth reading here.

xxvi PREFACE

On the other hand, we include as potential readers those who

have only studied some algebra.

1

For that reason we have explained

many topics, such as complex numbers and modular arithmetic,

which will be known to more advanced readers. In the later

chapters, where we discuss more abstruse topics, we sometimes

pause to explain things to readers with a more limited background.

This book follows a path to one particular area of modern number

theory: generalized reciprocity laws. It will take most of the book to

explain this concept. But here is an extremely brief description of

the territory:

We want to solve polynomial equations. Unlike the quadratic

equation ax

2

+ bx + c = 0, most equations do not have formu-

las that give you the solutions. Generalized reciprocity laws

are very complicated algorithms that enable you to get crucial

information about some of these more complicated equations.

In favorable circumstances, they can be used to prove deep

statements about the solutions sets of algebraic equations.

By the way, an equation doesn’t have to be too complicated to lack

a formula and require—if possible—a reciprocity law. For example,

x

5

+ ax + b = 0 lacks a formula. This surprising fact was proved in

the early 1800s by the Italian mathematician Paolo Rufﬁni and

the Norwegian mathematician Niels Henrik Abel and led to the

concept of the Galois group. One way to get mileage out of the

Galois group is to represent it in terms of either permutations or

matrices. These representations encapsulate the patterns referred

to in the subtitle of our book. From the representations we go on

to construct reciprocity laws. All of these terms—equation, group,

Galois group, permutation, matrix, representation, reciprocity law,

and many others—are deﬁned and discussed in the course of this

volume.

We can now explain the meaning of our title, although in a sense,

the full explanation requires the whole book. Some number pat-

terns, like even and odd numbers, lie on the surface. But the more

1

The few bits that mention trigonometry, logarithms, inﬁnite series, or differentiation

can be skipped without impairing your ability to follow the rest of the book.

Start Free Trial

No credit card required