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Fearless Symmetry by Robert Gross, Avner Ash

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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.

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