# Fearless Symmetry: Exposing the Hidden Patterns of Numbers (New Edition)

Avner Ash
Robert Gross
Pages: 312
https://www.jstor.org/stable/j.ctt7s74r

1. Front Matter
(pp. i-viii)
(pp. ix-xiv)
3. Foreword
(pp. xv-xx)
Barry Mazur

At some point in his or her life every working mathematician has to explain to someone, usually a relative, that mathematics is hardly a finished project. Mathematicians know, of course, that it is far too soon to put the glorious achievements of their trade into a big museum and just become happy curators. In many respects, the study of mathematics has hardly begun. But, at least in the past, this has not always been universally acknowledged.

Recent successes (most prominently the proof of Fermat’s Last Theorem) have advertised to a wide audience that math remains humanity’s grand “work-in-progress,” where mysteries...

4. Preface to the Paperback Edition
(pp. xxi-xxiv)
5. Preface
(pp. xxv-xxx)
6. Acknowledgments
(pp. xxxi-xxxii)
7. Greek Alphabet
(pp. xxxiii-xxxiv)
8. PART ONE. ALGEBRAIC PRELIMINARIES
• Chapter 1 REPRESENTATIONS
(pp. 3-12)

Before we narrow our focus to mathematical concepts, we start by discussing the general concept of a representation. In philosophy, the concept of one thing representing or misrepresenting another thing is a central concern. The distinction between truth and appearance, the thing-in-itself and its representation, is a keynote of philosophy. It plays a critical role in the works of such figures as Plato, Kant, Schopenhauer, and Nietzsche. Generally speaking, for these philosophers the “appearance” of something is thought to be an impediment or veil, which we wish to penetrate through to the reality acting behind it. But in mathematics, matters...

• Chapter 2 GROUPS
(pp. 13-20)

A group is a set along with a rule that tells how to combine any two elements in the set to get another element in the set. We usually use the wordcompositionto describe the act of combining two elements of the group to get a third.

We start our consideration of groups by thinking about a beautiful perfect sphere, one foot in radius, made of pure marble. Let it rest in a spherical container so it just fits exactly. Assume that we have a perfect map of the earth drawn on the sphere, so we can refer to...

• Chapter 3 PERMUTATIONS
(pp. 21-30)

Our next type of group goes back to the idea of one-to-one correspondence.¹ We start with a finite set, for example,${\rm{\{ a, b, c\} }}$. One thing they do in elementary school is to figure out all the possible ways of ordering this set. There are six ways:$a, b, c; a, c, b; b, a, c; b, c, a; c, a, b; c, b, a$. We can view any one of these orderings as the result of a one-to-one correspondence of this set with itself. For example, the third ordering can be viewed as the result of

$a \to b$

$b \to a$

$c \to c.$...

• Chapter 4 MODULAR ARITHMETIC
(pp. 31-41)

Modular arithmetic was invented and given that name by the nineteenth-century German mathematician Carl Friedrich Gauss, but the basic concept must be far older. It washed into American grade schools on the wave called “The New Math” in the 1950s and 1960s, and may still be found in some schools. The basic idea was usually illustrated by a problem of the following sort:

EXERCISE: Today is Tuesday. What day of the week will it be 25 days from now?

The student is supposed to realize that there are 7 days in a week: She can write 25 as 7 + 7 + 7 + 4, and...

• Chapter 5 COMPLEX NUMBERS
(pp. 42-48)

We do not absolutely need to use complex numbers to solve polynomial equations with integer coefficients. Instead, we can use a complicated algebraic recipe for inventing solutions as we need them and then keep track of them as we continue to add more solutions. If we wish to ignore the complex numbers, we can simply assume the existence of a large number system that contains all the solutions to all equations of the form$f(x) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + \cdots + {a_1}x + {a_0} = 0,$where the coefficients${a_n},{a_{n - 1}}, \ldots ,{a_1},{a_0}$are integers. In this case, you must remember that the same root can occur for many different polynomials. A sophisticated method...

• Chapter 6 EQUATIONS AND VARIETIES
(pp. 49-66)

Anequationis a statement, or assertion, that one thing is identical to another. For example, “The first president of the United States was George Washington.” Equality is timeless: Although we wrote “was,” because both sides of the equation existed in the past, logically speaking, “is” or “will be” have the same force as “was.”

Philosophically, equality can get pretty complicated. Consider the definition: “A unicorn is a horse with a single horn.” A unicorn and a single-horned horse are identical by definition even though they (it) do not exist. Yet “is” might imply existence.

To rid ourselves of this...

• Chapter 7 QUADRATIC RECIPROCITY
(pp. 67-84)

Let us discuss the solutions of polynomial equations. Since the time of the French mathematician and philosopher René Descartes, mathematicians have realized that thedegreeof a polynomial is a good indication of how complicated it is. Recall that the degree of a polynomial$f(x)$of one variable is the highest power of$x$that appears. For example,$10{x^{33}} - 100{x^7} + 1,729$has degree 33.

A polynomial of degree 0 is simply a constant. Because a rose is a rose is a rose, there is not much to say about a single constant, and we can go on to the next case.

A...

9. PART TWO. GALOIS THEORY AND REPRESENTATIONS
• Chapter 8 GALOIS THEORY
(pp. 87-102)

We review here the basic facts and definitions about polynomials.

1. A Z-polynomial is a polynomial in one variable with integer coefficients.

2. Thedegreeof a Z-polynomial is the degree of the highest power of the variable that occurs. For example, the degree of$11{x^5} + 103x - 41$is 5.

3. Any Z-polynomial$f$with degree at least 1 can be factored into a nonzero constant$c$times the product of factors of the form$(x - a),$where$a$is some complex number.

4. If the degree of$f$is$d$, then there are exactly$d$factors.

5. Therootsof$f$are...

• Chapter 9 ELLIPTIC CURVES
(pp. 103-113)

Serge Lang began his bookElliptic Curves: Diophantine Analysisby writing: “It is possible to write endlessly about elliptic curves. (This is not a threat.)” That is because an elliptic curve is simultaneously an example of two different concepts that we have discussed in earlier chapters: varieties and groups. An elliptic curve$E$is the set of solutions to a certain kind of Z-equation so that for any field$R$, that is, any number system in which we can divide by any nonzero element,$E\left( R \right)$is a group.¹

Moreover,$E\left( R \right)$is always anabelian group, which is the term that mathematicians...

• Chapter 10 MATRICES
(pp. 114-123)

We need matrices in order to get to the center of our subject—representations of Galois groups. There are two types of representations we must consider: matrix representations and permutation representations. We will discuss matrix representations in chapter 12 and permutation representations in chapter 14.

The two kinds of representations are closely related. Because we defined the Galois group as a group of permutations, one may think that it would be enough to study permutation representations. But even if we knew everything about the permutation representations, we would need the matrix kind in order to formulate some of the deeper...

• Chapter 11 GROUPS OF MATRICES
(pp. 124-134)

From now on, we will assume that the set of entries in our matrices consists of elements of a number system, which we will call$R$. This means that we can add, subtract, and multiply the elements of$R$, and they obey the usual laws of arithmetic. We will not need to use division at all, and so letting$R = {\bf{Z}},$for example, is fine.

The entries need not be actual numbers of the ordinary sort. For example, we could use the field${{\bf{F}}_2}$consisting of 0 and 1. Let us review a part of chapter 4. We define addition as...

• Chapter 12 GROUP REPRESENTATIONS
(pp. 135-148)

At last, we come to the heart of this book, or at least the pericardium. Agroup representationis nothing more nor less than a morphism from one group to another group. The reason we use the special term “representation” is that the target group is chosen to be one we are especially comfortable with, or one whose properties are important for understanding the source group. The two types of representations we will look at arepermutation representationsandlinear representations. What is a morphism from the group$H$to the group$K$? It is a function, call it$f\left( x \right)$,...

• Chapter 13 THE GALOIS GROUP OF A POLYNOMIAL
(pp. 149-156)

Recall that${Q^{alg}}$is the set of all roots of all Z-polynomials. We have called the group of all permutations of${Q^{alg}}$that preserve addition, subtraction, multiplication, and division the “absolute Galois group of$Q$.” (See chapter 8.) We have agreed to designate it by the letter$G$. It is a very large—in fact, infinite—group. In order to get a grasp on it, we are going to discuss a lot of smaller groups, also called Galois groups.

Each of these groups is the Galois group of a$Z$-polynomial. Here is the idea. Pick a$Z$-polynomial$f(x)$. Instead of...

• Chapter 14 THE RESTRICTION MORPHISM
(pp. 157-161)

We have now considered several examples of Galois groups of single$Z$-polynomials. Mathematicians like to put things together into larger things whenever possible. This activity is related to the problem of understanding the universe. Are all the different parts of the universe completely different from one another, or are they connected by various relationships, especially by cause and effect? How strong are these connections? Can you go so far as to say that the whole universe, and every detail in it, is the expression of a “One,” a single entity that somehow causes everything else? Unlike philosophers, mathematicians to some...

• Chapter 15 THE GREEKS HAD A NAME FOR IT
(pp. 162-176)

There are some amazing facts about finite groups and their linear representations. To explain them, we first need to talk about thetraceof a square matrix.

DEFINITION: Thetraceof a square matrix is the sum of the diagonal elements.

The “diagonal elements” are those elements that go from the upper left-hand corner to the lower right-hand corner of a square matrix.

EXAMPLE: The trace of

$\left[ {\begin{array}{*{20}{c}} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & {10} & {11} & {12} \\ {13} & {14} & {15} & {16} \\ \end{array} } \right]$

is 1 + 6 + 11 + 16 = 34.

We apply this apparently innocuous definition to matrix representations of groups.

DEFINITION: For any matrix representation$r$of the finite group$H$,...

• Chapter 16 FROBENIUS
(pp. 177-190)

Ferdinand Georg Frobenius was the nineteenth-century German mathematician who invented the method of using characters to study group representations.¹ Today, his name is used for (among other things) particular elements in the absolute Galois group$G$, and also their restriction to the Galois groups of various polynomials, that is, for particular permutations of the roots of various$Z$-polynomials.

Whenever you can get something for nothing in mathematics, you take it. Of course, “nothing” is a relative term. What we mean in this case is that we can define certain elements of every Galois group by using an easily appliedgeneral...

10. PART THREE. RECIPROCITY LAWS
• Chapter 17 RECIPROCITY LAWS
(pp. 193-199)

A little review: We have the set of all algebraic numbers, denoted${Q^{{\rm{alg}}}}$. We have the absolute Galois group$G$of all arithmeticpreserving permutations of${Q^{{\rm{alg}}}}$. If$R$is some number system,$n \ge 1$is some integer and$\phi :G \to {\rm{GL(}}n,R)$is a morphism of groups, then we say that$\phi$is a linear Galois representation with coefficients in$R$.

We have told you that each$\phi$comes with a set of “bad primes,” the “ramified primes.” In all of the examples we will consider, this set of bad primes is finite. If$p$is not in this finite list of bad primes,...

• Chapter 18 ONE- AND TWO-DIMENSIONAL REPRESENTATIONS
(pp. 200-215)

We start with the polynomial${x^n} - 1.$We know what the roots of this polynomial are, usingde Moivre’s Theorem: They are the$n$complex numbers$\zeta ,{\zeta ^2},{\zeta ^3}, \ldots ,{\zeta ^n},$where$\zeta = {\rm{cos}}\left( {{\textstyle{{2\pi } \over n}}} \right) + i{\rm{sin}}\left( {{\textstyle{{2\pi } \over n}}} \right).$It is important to know that these$n$numbers are all unequal, that they all solve the same polynomial equation${x^n} - 1 = 0,$and that${\zeta ^n} = 1.$One consequence of these facts is that$a \equiv b$(mod$n$) if and only if${\zeta ^a} = {\zeta ^b}.$¹

We look now at what the elements of the absolute Galois group$G$can do to these numbers. A key point is that when we know what$$\sigma$$, an element of$G$, does...

• Chapter 19 QUADRATIC RECIPROCITY REVISITED
(pp. 216-224)

Let us return to one-dimensional representations, and show how we can use the concept of a generalized reciprocity law to deduce quadratic reciprocity. The black box associated with the Galois representations we consider in this chapter is constructed from an integer$N > 1$and a set of functions${F_N}$with some special properties. First we have to make the following definition:

DEFINITION: Let$N$be an integer greater than 1. Then${({\bf{Z}}{\rm{/}}N{\bf{Z}})^ \times }$is the set of all integers$b$modulo$N$that have multiplicative inverses modulo$N$.

For example, 3 is in${({\bf{Z}}{\rm{/10}}{\bf{Z}}{\rm{)}}^ \times }$because$3 \cdot 7 \equiv 1$(mod 10), but 5 is not...

• Chapter 20 A MACHINE FOR MAKING GALOIS REPRESENTATIONS
(pp. 225-232)

We have seen in preceding chapters examples of one- and twodimensional Galois representations. But it is very unlikely that they include all the information to be found in the absolute Galois group of Q. Indeed they do not. We seek a large supply of Galois representations of all dimensions.

We need a general method to derive linear Galois representations from$Z$-varieties. There are several of these methods, but we discuss just one:étale cohomology. It is too advanced to explain in detail here, but we can give a very rough idea of what is going on. First, a definition:

DEFINITION:...

• Chapter 21 A LAST LOOK AT RECIPROCITY
(pp. 233-241)

A fair definition of mathematics might be “the logical study of patterns”:

Patterns of numbers—look at the chapters on quadratic reciprocity for some complicated but beautiful examples.

Patterns of permutations—for example, the behavior of cycle decompositions.

Patterns of points—this is geometry.

Patterns of solutions to systems of$Z$-equations—this is what this book is all about.

We use the term “pattern” in the broadest possible sense, to mean any arrangement of things that follows some orderly rule, allowing for prediction or contemplation. A simple pattern is the empty set.¹ If you know the...

• Chapter 22 FERMAT’S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS
(pp. 242-256)

In the last chapter, we gave our last look at reciprocity laws per se. Now we will show how these laws can be applied to prove Fermat’s Last Theorem (referred to as FLT throughout this chapter) and similar theorems.

Before Andrew Wiles burst upon the scene, there were a number of proposed strategies for proving FLT. Let us begin by explaining the strategy that Wiles ultimately made to work. This strategy is composed of several difficult theorems plus one big conjecture. The proof was completed by Wiles and Taylor–Wiles by proving enough of the big conjecture to make the...

• Chapter 23 RETROSPECT
(pp. 257-264)

Now that we have been over the sometimes rocky, sometimes (we hope) beguiling road of this book—there being no Royal Road to mathematics—it is time to look back and see where we have been, and also to take a glimpse of the view in front of us. Our goal was to give you some idea of the absolute Galois group of$Q$and its representations. We defined and explained basic mathematical concepts needed before we could even begin our discussion: sets, groups, various number systems, matrices, and representations. These concepts are not confined to number theory. They are...

11. Bibliography
(pp. 265-268)
12. Index
(pp. 269-272)