92 CHAPTER 8
the limit, you get the value of
. Because we have to perform a
limiting process, this innocuous expression
really belongs to
calculus and not to algebra.
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
. 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.
DEFINITION: The absolute Galois group G is made up of all
permutations g of Q
that preserve addition and
multiplication. That is, for any numbers a and b in Q
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
is an inﬁnite set, its group of
permutations is also inﬁnite. But we do not let just any permutation
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
. The point
is that there are ways to deﬁne
2 without resort to a limiting process, namely, as the
positive root of x
− 2, while a difﬁcult theorem asserts that there is no way to deﬁne
without using a limiting process.
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.