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Fearless Symmetry by Robert Gross, Avner Ash

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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.

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