THE RESTRICTION MORPHISM 159

roots of f (x) and on the ﬁeld they generate, forgetting about what γ

may be doing to other algebraic numbers. When we do this, we see

that γ has turned into an element of G(f ). In other words, there is

an element σ of G(f ) that is speciﬁed by saying: σ does exactly what

γ does in the ﬁeld of numbers Q(f ) generated by the roots of f (x).

This gives us a function from G to G(f ), which sends γ to σ .

Because the group law in both G and G(f ) is just composition of

permutations, this function preserves the group law. In other words,

it is a morphism. It is called the restriction morphism, and we will

call it r

G( f )

.IfyoustartwithanewZ-polynomial, say h(x), and get

a new Galois group G(h) that acts in the ﬁeld Q(h) generated by

its roots, then we get another restriction morphism, and we will

call it r

G(h)

. And so on. Thus, we have inﬁnitely many restriction

morphisms, r

G( f )

: G → G(f ), r

G(h)

: G → G(h), and so on.

Basic Facts about the Restriction Morphism

In this way, we obtain a lot of restriction morphisms. Every Galois

group of a Z-polynomial is the target of exactly one of them.

A big theorem in Field Theory tells us that every one of these

restriction morphisms is “surjective.”

1

This means that if we take

any particular Z-polynomial f , and any allowable permutation σ of

the roots of f (allowable in the sense that it is in G(f )), then we can

ﬁnd some element γ in G that does exactly the same permutation of

the roots of this particular polynomial while doing who-knows-what

to all of the other algebraic numbers—that is, r

G( f )

(γ ) = σ .

So G really does contain all the information of all the different

G(f )’s . You get back information about G(f ) by applying r

G( f )

to

whatever you might know about G. Therefore, G is like a Holy

Grail. If we understood G completely, we would know a lot about

all the different Z-polynomials and their Galois groups.

A bonus of wrapping all the information up into G is that

certain theorems become easier to state or easier to remember.

It helps to have the single object G as our focus. Also, we can

1

This is the “existence theorem” that we mentioned back in chapter 8.

Start Free Trial

No credit card required