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
, we only look at the part we get by taking all of the roots of
f (x) (all contained in Q
) 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).
EXAMPLE: If f (x) is the polynomial (x
− 7), some of
the numbers you get in the pot after enough stirring will be
, 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
are also in the ﬁeld (in the case of
, we assume that v = 0).
Secondly, Q(f )isthesmallest ﬁeld that contains all the roots of f (x)
and all rational numbers.
Now we have the ﬁeld Q(f ), which is a subset of Q
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
A theorem tells us that the contents actually will stop growing after some ﬁnite number
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
= 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
to get all of the rational numbers.