We are pleased by the reception received by the hardcover edition

of Fearless Symmetry. Our readers include high-school students

and particle physicists. Some people read the ﬁrst few chapters

with enthusiasm and then put the book aside. We hope they will

return to the later chapters in the future. Other readers, with more

background, can skip the ﬁrst part of the book and dig deeper into

the absolute Galois group and the topics that follow. The climactic

chapter on Fermat’s Last Theorem is very difﬁcult, but we hope

that, even read cursorily, it gives a more realistic view of what was

at stake in the proof of that theorem than do the more journalistic

treatments that are available.

Many people contacted us with minor corrections, questions, and

comments. One correspondent even sent us an entire dossier ﬁlled

with Mathematica programs designed to do some of the exercises.

We have inserted the minor corrections into this paperback

edition. However, we have been unable to use a number of excel-

lent suggestions for improving our exposition because of technical

constraints. We take the opportunity of this new preface to reply

to four interesting observations from our correspondents about

speciﬁc points in the book.

On page 14, footnote 2, we deﬁned “counterclockwise” in terms

of unscrewing a light bulb while looking down at it. A reader

from Australia complained. At ﬁrst, we were afraid that we were

guilty of Americo-centrism, and that in Australia, light bulbs were

unscrewed clockwise. But it turned out that the complaint was that

xxii PREFACE TO THE PAPERBACK EDITION

one is usually looking up at a ceiling ﬁxture when unscrewing

a light bulb. Of course, we were thinking of a table lamp. But

this exchange did lead to the question of whether there do exist

light bulbs with reverse threads. According to several sites on the

Internet, the Metropolitan Transit Authority in New York City did

employ such bulbs, presumably in order to prevent theft.

Most amusing was something that happened on page 58. In the

exercise, we wrote down at random what seemed to us a sufﬁciently

complicated system of equations to make it very hard or impossible

to ﬁnd its integral solutions. Yet one of our correspondents wrote

to us with a complete solution to this exercise! The odd thing is

that our attempt in an earlier draft to write down a not-too-lengthy

but “hard” system of equations had resulted in a system that we

rejected when we realized that it could in fact be solved. In other

words, we are terrible at writing down random hard systems of

equations. Our correspondent suggested that there could be a deep

property of human beings that makes it difﬁcult for us to write

down short “unsolvable” random-looking algebraic systems.

In the discussion on pages 97 and 98, we said that no element

in the absolute Galois group of Q can be explicitly described except

for the identity element and complex conjugation. We stand by this

statement. But one of our correspondents didn’t believe us and

bravely attempted to describe a third element by specifying what

it did to various roots of integers. Other readers may have made

similar attempts. Even if this could be done consistently for all roots

of integers, it wouldn’t be enough, because the ﬁeld generated by

all roots of integers is far from being the entire ﬁeld of algebraic

numbers Q

alg

. Of course, if you have inﬁnite time and an inﬁnite

amount of paper, then you can explicitly write down other elements

in the absolute Galois group.

Finally, on page 179, we say that “we are never going to have

a naked Frob

p

in any formula,” but in a number of places we

seem to have just that. For example, on page 185, we state that

Frob

3

(i) =−i. The nudity of Frob

p

is potentially a problem because

there is a certain ambiguity in the notation “Frob

p

.” However, when

we evaluate Frob

p

at the number i, when p is odd, the ambiguity

goes away, the same way it does when we consider the character

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