92 CHAPTER 8

the limit, you get the value of

√

2

√

3

. Because we have to perform a

limiting process, this innocuous expression

√

2

√

3

really belongs to

calculus and not to algebra.

2

The Absolute Galois Group of Q Deﬁned

How do we bring order to what seems to be a chaos of algebraic

numbers? One could paraphrase Alexander Pope’s couplet on Isaac

Newton and say: God said, “Let Galois be.” But that could be

overstating the case. There is still a lot of chaos left over, even after

we understand Galois theory. However, Galois theory is what gives

us a way to ask intelligent questions about this swirling mass of

algebraic numbers—that, together with representation theory.

We begin by deﬁning the “absolute Galois group of Q.” It is

usually denoted by the symbol G

Q

. But for short, we will call it “the

Galois group” and usually denote it simply by the letter G.Itis

called “absolute” to distinguish it from the various “relative” Galois

groups of polynomials that will be introduced in chapter 13.

3

DEFINITION: The absolute Galois group G is made up of all

permutations g of Q

alg

that preserve addition and

multiplication. That is, for any numbers a and b in Q

alg

,we

must have g(a + b) = g(a) + g(b) and g(ab) = g(a)g(b).

Here is a brief dialogue that will help to explain this deﬁnition.

First, notice that because Q

alg

is an inﬁnite set, its group of

permutations is also inﬁnite. But we do not let just any permutation

of Q

alg

into G.

2

The reader might not be satisﬁed with this discussion. After all,

√

2 can be deﬁned by

a limiting process as well, and yet we are allowing

√

2 as an element of Q

alg

. The point

is that there are ways to deﬁne

√

2 without resort to a limiting process, namely, as the

positive root of x

2

− 2, while a difﬁcult theorem asserts that there is no way to deﬁne

√

2

√

3

without using a limiting process.

3

This would be a good time to review the concept of “permutation” from chapter 3, if you

need to. Remember that a permutation of a set A is a function from A to itself, which is

a one-to-one correspondence.

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