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
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
I give you an element a of Q
, 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?
—Good. And would you ever give the same output for two
different algebraic number inputs?
—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
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