GALOIS THEORY 93

A Conversation with s: A Playlet in Three Short Scenes

Meet the permutation s:

—Hello. Pleased to meet you. So you are a one-to-one

correspondence between Q

alg

and itself? And you would like

to join the G-club? I’m afraid there are some tests you will

have to pass before we can admit you.

—Please, I’m ready to try.

—First, let’s check that you really are a permutation of Q

alg

.If

I give you an element a of Q

alg

, that is to say, if I give you

an algebraic number a, what will you do with it?

—I’ll take it and output another algebraic number, let’s say b.

Only one output. I’m never in doubt. And I always turn a

into b. I’m an honest function.

—Are you sure you won’t ever output π or something

nonalgebraic like that?

—I won’t.

—Good. And would you ever give the same output for two

different algebraic number inputs?

—No.

—That’s good too. And might there be some algebraic number

that is never an output of yours?

—No. If you name any algebraic number at all, for example

√

2 +

5

√

11, then I can ﬁnd some algebraic number input

whose output it would be.

—Very well, so you are a permutation. But that’s still very far

from what you need to enter G. So far, all I’ve done is

checked your status as a permutation. The big test lies

ahead. Let’s go slowly. What do you do with 0 as input?

—I output 0.

—Good. What do you do with 1?

—I output 1.

(This could go on quite a while, so we will summarize a bit.

We’ll say that s “sends” a to b if for the input a she outputs b.

In functional notation we could write b = s(a). It turns out

that s sends every rational number, that is, every fraction

c

d

,

94 CHAPTER 8

where c and d are integers, to itself. That is good, because

we will prove later in this chapter that s must have this

property in order to belong to the Galois group G.)

—Maybe you are just the identity permutation. Do you in fact

send every algebraic number to itself?

—No, I’m more interesting than that.

—Good, glad to hear that. The identity permutation is

already a member of the G-club anyway. Now let’s get down

to business. What do you do with

√

2?

—I send it to −

√

2.

❆❆

(Later that afternoon.)

—What do you do with a root of 1 + x

2

+ 33x

3

− x

7

+ 4x

9

?

—I send it to another root of the same polynomial

1 + x

2

+ 33x

3

− x

7

+ 4x

9

.

—Hmm. Is that a general rule? If I give you a root of any

Z-polynomial, will you always send it to another root,

either the same one or a different one, of the same

Z-polynomial?

—(proudly) Yes, I will.

—Good, things look hopeful. But now we have to come down

to brass tacks. Do you preserve all arithmetic operations:

addition, subtraction, multiplication, and division?

—Excuse me, I’m not sure what you mean.

—Well, for example, suppose you send a to c and b to d. Tell

me what you will do with a + b.

—I’ll send it to c + d.

—Are you sure? Always?

—Yes. For example, because I send

√

3 to itself—

√

3—and I

send

√

2to−

√

2, then I will send

√

3 +

√

2to...let me

check ...yes, to

√

3 −

√

2, as I claimed. Also, because I send

5 to itself and

√

2to−

√

2, then I will send 5 +

√

2to

5 −

√

2. And so on.

—What about a − b, ab and

a

b

?

—I’ll send them to c − d, cd and

c

d

, without fail.

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