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Fearless Symmetry by Robert Gross, Avner Ash

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We patch together the Galois groups of polynomials into
the absolute Galois group G via a type of morphism
of groups called a “restriction morphism. This glues
together all the G(f )’s and conversely gives us a method
for studying G by looking at the G(f )’s. It also provides us
with a lot of permutation representations of G.
If we somehow learn something about G, we can ap-
ply the restriction morphism to get information about a
particular G(f ) in which we may be interested. Later,
we will replace G(f ) with even more complicated objects
related to Z-varieties. In this way, we can prove results
about varieties such as Fermat’s Last Theorem, or, more
generally, theorems about Diophantine equations such as
x
p
+ y
q
= z
r
.
The Big Picture and the Little Pictures
We have now considered several examples of Galois groups of single
Z-polynomials. Mathematicians like to put things together into
larger things whenever possible. This activity is related to the
problem of understanding the universe. Are all the different parts
of the universe completely different from one another, or are they
connected by various relationships, especially by cause and effect?
How strong are these connections? Can you go so far as to say
158 CHAPTER 14
that the whole universe, and every detail in it, is the expression
of a “One, a single entity that somehow causes everything else?
Unlike philosophers, mathematicians to some extent get to invent
or arrange their universe the way they like. It is certainly useful
to group similar things together. Sometimes, this causes one large
object to form, which becomes a useful tool for looking at all the
individual things. This is one way to look at G, the absolute Galois
group of Q.
What is the connection between G and the Galois group G(f )
of a given polynomial f (x)? There is a morphism from G to G(f )
called the restriction morphism. It is the crucial glue that holds all
the possible G(f )’s together and welds them into G,or,conversely,
that allows us to go from G, which is an infinite group, to the more
understandable realm of the individual finite G(f )’s.
The world is nothing more than all the individuals and neigh-
borhoods and countries that comprise it, fitted together with in-
numerable relationships of different kinds. In mathematics, the
relationships are less varied and more mechanical. This difference
makes mathematics easier, or at least more reliable, than politics.
The basic relationship between G and G(f ), the restriction mor-
phism, is a sort of passport that guarantees that G(f ) fits into G.
Here is how to describe the restriction morphism: Remember
that Q
alg
is the set of all those complex numbers that are the roots of
Z-polynomials. And G is the group of all one-to-one correspondences
of Q
alg
with itself that preserve all the arithmetic operations of
addition, subtraction, multiplication, and division. What is G(f )?
It consists of those permutations of the roots of f (x) that preserve
the arithmetic operations in the field of numbers Q(f ) generated by
those roots.
Now, as we saw in chapter 8, any element γ in G will permute the
roots of any given Z-polynomial. Of course, γ is very busy permut-
ing all the roots of all the Z-polynomials at the same time . But if we
focus in on f (x), we see that, in particular, γ is permuting its roots.
Of course, because γ preserves all the arithmetic operations among
all the algebraic numbers, in particular γ preserves the arithmetic
operations in the field of numbers generated by the roots of f (x). In
other words, we can restrict our attention to see how γ acts on the

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