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Fearless Symmetry by Robert Gross, Avner Ash

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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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