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Fearless Symmetry by Robert Gross, Avner Ash

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96 CHAPTER 8
Digression: Symmetry
A symmetry is a function that preserves what we feel is important
about an object. For example, a starfish has fivefold symmetry,
which means there is a function (rotation by 72
) that we can
perform on the starfish which keeps it looking pretty much the
same. (Fivefold means that if we perform this function five times
we will come back to the original state.) Humans have twofold
symmetry, because if we reflect them in a mirror, they still look
pretty much the same. If we rotate humans 180
, they will be
standing on their heads or facing away from the mirror (or both). So
reflection in a mirror is a different kind of symmetry from rotation.
But it is twofold because if we reflect the reflection we get back the
original.
These symmetries are not exact, for nothing in living nature is
absolutely exact. But in mathematics we can study exact symme-
tries.
5
In our case, each g in the Galois group G is a symmetry
of Q
alg
because it preserves the operations that concern us in
algebra, namely, addition and multiplication. This use of the word
“symmetry” is part of the justification for the title of this book.
How Elements of G Behave
The four equations (8.3)–(8.6) might seem like no big deal, but
they have many consequences and put many constraints on the
permutation g if it is to be allowed into G. Some of these constraints
are obvious and some of them are very subtle—so subtle that Galois
theory still possesses a large realm of mystery.
Let us look at some of the less subtle constraints on g. First, if
we set a = b, then from equation (8.4) we derive g(0) = 0, and if we
set a = b = 1, we can then derive from equation (8.5) that g(1) = 1.
6
So g must send 0 to 0 and 1 to 1. This explains the interviewer’s
first two questions.
5
A classic reference about symmetries is (Weyl, 1989).
6
Because g is a permutation and g(0) = 0, it follows that g(1) = 0, so it can be cancelled
on both sides of the equation g(1) = g(1)g(1).
GALOIS THEORY 97
Now let a = 1 and b = 1 and use equation (8.3). We obtain
g(2) = 2. So g must send 2 to 2. Do you see the pattern? Setting
a = 1 and b = 2, we can now use equation (8.3) again to see that g
must send 3 to 3. And so on. We conclude that if a is any positive
integer, then g(a) = a.
Notice that this argument is a kind of bootstrap. We cannot show
directly that g(13) = 13, but we have to build it up slowly, first
1, then 2, then 3, and so on, until we get to 13. Slow but sure
wins the race—we can eventually bootstrap ourselves up to any
positive integer we want, and so we have justified the statement
that g(a) = a for every positive integer a. This kind of argument
has a formal name: mathematical induction.
EXERCISE: Use equation (8.4) over and over again, starting
with a = 0 and b = 1, to show that g(k) = k for every negative
integer k.
So we have seen that if g is in G, then it takes every integer
to itself. But now if we have the fraction
a
b
, where b is not zero,
and a and b are integers, we can use formula (8.6) and derive
that g(
a
b
) =
g(a)
g(b)
=
a
b
. In other words, g takes every rational number
to itself. This is why the interviewer kept asking this type of
question.
Perhaps you are beginning to suspect that g must take every
algebraic number to itself, so that G would contain only one
element, the identity permutation, which we shall call e. In this
case, G would not be very interesting and we would not be talking
about it. But because you have met s, you know that there are other
elements in G. Our acquaintance s, for example, takes
2to
2.
In fact, G has infinitely many elements.
You may want us to describe in some thorough way an element in
G that is not the identity permutation. But here is an amazing fact,
one of the facts that make Galois theory so hard: There is only one
element of G, other than the neutral element e, for which we can
give a complete description that would satisfy you. This element is
called c and we will describe it in the next paragraph. But except
for e and c, no other element of G can be written down explicitly.

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