260 CHAPTER 23
each prime p not dividing the level, the number of solutions
). For a survey of this and many similar examples, see
Digression: Why Do Math?
We can attempt an answer to the question “Why do math?” as
illustrated by the mathematics discussed in this book.
If mathematics is the study of patterns, then we immediately
see the two reasons throughout history why people have “done
1. Curiosity about patterns, particularly of numbers and
shapes, and an æsthetic appreciation of them.
2. The need to study patterns occurring in the “real world.”
People sufﬁciently motivated by the ﬁrst reason become “pure”
mathematicians. They generally cannot understand why everyone
does not share their enthusiasm for the beauty of these patterns.
People motivated by the second reason are everyone (including
also “pure” mathematicians). Even in mundane affairs, you cannot
ignore patterns, such as those in your checkbook or the number
patterns in the bus schedule; the arrangement of the rooms and
furniture in your apartment (spatial patterns); the knotting of
DNA (topological patterns) if you are a molecular biologist, “etc.
We see in these two reasons the sources of “pure” and “applied”
mathematics, respectively. This book has concerned itself with a
topic in pure mathematics, a subtopic of number theory. Number
theory is often thought of as the purest of mathematics. Yet even the
ideas in this book, such as prime numbers,
etale cohomology, and
the corresponding Galois representations, have been ﬁnding their
applications in the “real world.” For example, Andr
e Weil’s version
of the Riemann Hypothesis for curves over F
has been applied
to the problem of constructing efﬁcient communication networks.
The use of prime numbers and their theory in the construction
of public key codes is of ever-increasing importance for ﬁnancial