THE RESTRICTION MORPHISM 161

same thing to every root of every Z-polynomial. So γ and γ

are the

same.

The importance of the restriction morphisms cannot be over-

stressed. The absolute Galois group is inﬁnite, and we humans

cannot grasp it all at once. All we see are its manifestations in our

human world, namely, how it acts, via the restriction morphisms,

on the various Galois groups of single Z-polynomials. At least

theoretically, we can understand any single Z-polynomial if we put

our minds to it. We cannot grasp G all at once. But by seeing what it

does to the roots of individual Z-polynomials, we can hope to prove

theorems about G.

In fact, we can sometimes do more. Sometimes there are whole

inﬁnite families of Z-polynomials we can study together. For

example, there is the family x

n

− 1, where n = 1, 2, 3, .... We will

look more closely at these in chapter 18 when we investigate one-

dimensional Galois representations.

Examples

We ﬁnish this chapter with a couple of examples. The only elements

in G that we can describe totally explicitly are the neutral element

ι and complex conjugation c. So we will give one example for each.

EXAMPLE: Whatever Z-polynomial f may be, r

G( f )

(ι)isalways

the identity permutation.

EXAMPLE: Refer to the example in the last chapter where

f (x) = x

3

− 5 and use the notation introduced there for its

three roots, r, s, t: r =

3

√

5, s = ω

3

√

5, and t = ω

2

3

√

5. Now,

c(

3

√

5) =

3

√

5because

3

√

5 is real. Also, c(ω) = ω

2

,andc(ω

2

) = ω

(which you can deduce from the fact that ω and ω

2

are the two

roots of x

2

+ x + 1, or you can work it out directly from the

value of ω given in the previous chapter).

So you see that r

G( f )

(c) performs the permutation r → r,

s → t,andt → s.

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