On the other hand, we include as potential readers those who
have only studied some algebra.
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
We want to solve polynomial equations. Unlike the quadratic
+ 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,
+ 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
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
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.