We patch together the Galois groups of polynomials into

the absolute Galois group G via a type of morphism

of groups called a “restriction morphism.” This glues

together all the G(f )’s and conversely gives us a method

for studying G by looking at the G(f )’s. It also provides us

with a lot of permutation representations of G.

If we somehow learn something about G, we can ap-

ply the restriction morphism to get information about a

particular G(f ) in which we may be interested. Later,

we will replace G(f ) with even more complicated objects

related to Z-varieties. In this way, we can prove results

about varieties such as Fermat’s Last Theorem, or, more

generally, theorems about Diophantine equations such as

x

p

+ y

q

= z

r

.

The Big Picture and the Little Pictures

We have now considered several examples of Galois groups of single

Z-polynomials. Mathematicians like to put things together into

larger things whenever possible. This activity is related to the

problem of understanding the universe. Are all the different parts

of the universe completely different from one another, or are they

connected by various relationships, especially by cause and effect?

How strong are these connections? Can you go so far as to say

158 CHAPTER 14

that the whole universe, and every detail in it, is the expression

of a “One,” a single entity that somehow causes everything else?

Unlike philosophers, mathematicians to some extent get to invent

or arrange their universe the way they like. It is certainly useful

to group similar things together. Sometimes, this causes one large

object to form, which becomes a useful tool for looking at all the

individual things. This is one way to look at G, the absolute Galois

group of Q.

What is the connection between G and the Galois group G(f )

of a given polynomial f (x)? There is a morphism from G to G(f )

called the restriction morphism. It is the crucial glue that holds all

the possible G(f )’s together and welds them into G,or,conversely,

that allows us to go from G, which is an inﬁnite group, to the more

understandable realm of the individual ﬁnite G(f )’s.

The world is nothing more than all the individuals and neigh-

borhoods and countries that comprise it, ﬁtted together with in-

numerable relationships of different kinds. In mathematics, the

relationships are less varied and more mechanical. This difference

makes mathematics easier, or at least more reliable, than politics.

The basic relationship between G and G(f ), the restriction mor-

phism, is a sort of passport that guarantees that G(f ) ﬁts into G.

Here is how to describe the restriction morphism: Remember

that Q

alg

is the set of all those complex numbers that are the roots of

Z-polynomials. And G is the group of all one-to-one correspondences

of Q

alg

with itself that preserve all the arithmetic operations of

addition, subtraction, multiplication, and division. What is G(f )?

It consists of those permutations of the roots of f (x) that preserve

the arithmetic operations in the ﬁeld of numbers Q(f ) generated by

those roots.

Now, as we saw in chapter 8, any element γ in G will permute the

roots of any given Z-polynomial. Of course, γ is very busy permut-

ing all the roots of all the Z-polynomials at the same time . But if we

focus in on f (x), we see that, in particular, γ is permuting its roots.

Of course, because γ preserves all the arithmetic operations among

all the algebraic numbers, in particular γ preserves the arithmetic

operations in the ﬁeld of numbers generated by the roots of f (x). In

other words, we can restrict our attention to see how γ acts on the

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