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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