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Fearless Symmetry by Robert Gross, Avner Ash

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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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