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Fearless Symmetry by Robert Gross, Avner Ash

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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
3i4
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 1toget0,01 1 −···−1to
get the rest of the integers, and then
a
b
to get all of the rational numbers.

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