The absolute Galois group G is very large and difﬁcult

to understand. We can glimpse various aspects of it by

constructing permutation representations of G that have

as a target some ﬁnite permutation group. It is like trying

to understand an inﬁnite universe by looking at photos of

small pieces of it—a nebula here, a supernova there. Fig-

uring out how these photos piece together is very difﬁcult.

The Reciprocity Laws to be discussed in chapter 17 are

one of the ways of obtaining a bigger picture.

In this chapter we discuss the individual “photos”

themselves. Each one is a Galois group of a Z-polynomial.

In the next chapter we will glue these onto G and hence

to each other via “the restriction morphisms.” We will see

that the inﬁnite set of all possible “photos” does cover the

whole of G.

The Field Generated by a Z-Polynomial

Recall that Q

alg

is the set of all roots of all Z-polynomials. We have

called the group of all permutations of Q

alg

that preserve addition,

subtraction, multiplication, and division the “absolute Galois group

of Q.” (See chapter 8.) We have agreed to designate it by the letter G.

It is a very large—in fact, inﬁnite—group. In order to get a grasp on

it, we are going to discuss a lot of smaller groups, also called Galois

groups.

150 CHAPTER 13

Each of these groups is the Galois group of a Z-polynomial. Here

is the idea. Pick a Z-polynomial f (x). Instead of looking at all of

Q

alg

, we only look at the part we get by taking all of the roots of

f (x) (all contained in Q

alg

) and all of their arithmetic combinations.

If f (x) has degree d, then it will have at most d roots. By making

arithmetic combinations of them, we get a whole bunch of numbers

called a “ﬁeld.” Follow this recipe:

Start with a big pot. Take a particular Z-polynomial f (x) of degree

d ≥ 1. Then throw into the pot all of the roots of f (x), along with all

rational numbers. Then stir. Stirring means we form all possible

sums, differences, products, and quotients of all the numbers in

the pot. Then stir again, and keep doing this over and over again,

watching the brew grow—double, double, toil, and trouble. After

stirring inﬁnitely many times, we call the result Q(f )—the ﬁeld

generated by the roots of f (x).

1

EXAMPLE: If f (x) is the polynomial (x

2

+ 1)(x

3

− 7), some of

the numbers you get in the pot after enough stirring will be

3

4

,

i,

3

√

7,

i+

3

√

7−

3

4

77

,

i

3

√

7+i

5

3

√

7

2

3i−4

3

√

7+8

, and so on.

There are other ways to describe Q(f ). First of all, it is a ﬁeld.

Recall the deﬁnition of a ﬁeld. It is a set of numbers closed under

addition, subtraction, multiplication, and division. This means that

if u and v are any two elements of the ﬁeld, then u + v, u − v, uv

and

u

v

are also in the ﬁeld (in the case of

u

v

, we assume that v = 0).

Secondly, Q(f )isthesmallest ﬁeld that contains all the roots of f (x)

and all rational numbers.

2

Now we have the ﬁeld Q(f ), which is a subset of Q

alg

.Wecan

look at the permutations of Q(f ) that preserve all the arithmetic

operations. This set of permutations is a group, called the Galois

group of f (x), denoted G(f ). The group G(f ) is always ﬁnite, which

1

A theorem tells us that the contents actually will stop growing after some ﬁnite number

of stirrings.

2

All rational numbers are in Q( f ) because by deﬁnition we threw them into the pot at

the start. However, it is interesting to note that if f (x) has at least one nonzero root,

say u, then we can build up all of the rationals from u alone by forming

u

u

= 1, then

1 + 1 +···+1 to get all of the positive integers, then 1 − 1toget0,0−1 − 1 −···−1to

get the rest of the integers, and then

a

b

to get all of the rational numbers.

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